Towards a classification of connected components of the strata of k-differentials
aa r X i v : . [ m a t h . G T ] J a n TOWARDS A CLASSIFICATION OF CONNECTED COMPONENTSOF THE STRATA OF k -DIFFERENTIALS DAWEI CHEN AND QUENTIN GENDRON
Abstract. A k -differential on a Riemann surface is a section of the k -th power of thecanonical bundle. Loci of k -differentials with prescribed number and multiplicities ofzeros and poles form a natural stratification for the moduli space of k -differentials.The classification of connected components of the strata of k -differentials was knownfor holomorphic differentials, meromorphic differentials and quadratic differentialswith at worst simple poles by Kontsevich–Zorich, Boissy and Lanneau, respectively.Built on their work we develop new techniques to study connected components ofthe strata of k -differentials for general k . As an application, we give a completeclassification of connected components of the strata of quadratic differentials witharbitrary poles. Moreover, we distinguish certain components of the strata of k -differentials by generalizing the hyperelliptic structure and spin parity for higher k .We also describe an approach to determine explicitly parities of k -differentials ingenus zero and one, which inspires an amusing conjecture in number theory. A keyviewpoint we use is the notion of multi-scale k -differentials introduced by Bainbridge–Chen–Gendron–Grushevsky–M¨oller for k = 1 and extended by Costantini–M¨oller–Zachhuber for all k . Contents
1. Introduction 12. Multi-scale and marked k -differentials 53. Basic operations on k -differentials 84. Hyperelliptic components of the strata of k -differentials 195. Parity of the strata of k -differentials 226. Adjacency of the strata of quadratic differentials 327. Quadratic differentials with metric poles 36Appendix A. Parity of k -differentials in genus zero and one 50References 591. Introduction
Let g ≥ k ≥ µ = ( k , . . . , k n ) be an integral partition of k (2 g − k -differentials Ω k M g ( µ ) parameterizes sections of the k -th Date : January 6, 2021.Research of D.C. was supported in part by NSF Standard Grant DMS-2001040, NSF CAREERGrant DMS-1350396, and Simons Foundation Collaboration Grant 635235.Research of Q.G. was supported in part by a Postdoctoral Fellowship from DGAPA, UNAM. power of the canonical bundle on genus g Riemann surfaces with n distinct zeros or polesof order specified by the signature µ . It is known that Ω k M g ( µ ) is a complex orbifold(see e.g. [Bai+19a]), but it can be disconnected for special µ . Thus to understandthe topology of Ω k M g ( µ ), an important question is the classification of its connectedcomponents.For abelian differentials (i.e. k = 1), connected components of the strata are classifiedby [KZ03] for holomorphic differentials and by [Boi15] for meromorphic differentials. Inthis case, there can exist extra components due to hyperelliptic and spin structures forcertain signatures µ . For quadratic differentials with at worst simple poles (i.e. k = 2and k i ≥ − µ ), connected components of the strata are classifiedby [Lan08]. In this case, extra components are caused by the hyperelliptic structureonly, except in genus three and four where several sporadical components exist due toa strange mod 3 parity [CM14]. Built on the strategies of these works, we develop inthis paper a framework and new techniques towards solving the remaining cases.Note that connected components of the strata of k -differentials are known in genuszero and one for all k . For g = 0 and µ = ( k , . . . , k n ), the (projectivized) stratum isisomorphic to the moduli space of n -pointed P , hence it is irreducible. For g = 1, sincethe canonical bundle is trivial on a torus, the stratum is isomorphic to the correspondingstratum of abelian differentials with the same signature, hence the result of [Boi15,Section 4] applies for all k (see Theorem 3.12). Therefore, we can concentrate on thecase of g ≥ k -differential is called primitive if it is not a power of a lower-order differential.By taking powers, connected components of the strata of lower-order differentials giveconnected components of the corresponding loci in the strata of higher-order differen-tials, hence we can further restrict our study to the loci of primitive k -differentials. Weuse Ω k M g ( µ ) prim to denote the primitive locus in the stratum Ω k M g ( µ ).The results of Kontsevich–Zorich, Boissy and Lanneau for abelian and quadraticdifferentials were proven by induction on the genus and on the number of singularities.Two important operations they used in this process are breaking up a zero into lower-order zeros and bubbling a handle at a zero so as to increase the genus by one. Inorder to extend these operations to k -differentials for general k , we use as a key toolthe theory of multi-scale k -differentials, which was introduced for abelian differentialsin [Bai+19b] and extended for all k in [CMZ19]. The moduli space of multi-scale k -differentials provides a smooth and functorial compactification for the strata of k -differentials. In particular, smoothing a multi-scale k -differential from the boundary ofa stratum into the interior singles out a unique connected component of the stratum, forotherwise different components of the stratum would intersect in the boundary whichviolates the smoothness of the moduli space of multi-scale k -differentials. Using thisprinciple together with other techniques, we obtain a number of results towards theclassification of connected components of the strata as follows.We first generalize the hyperelliptic structure from the case of k ≤ k . Theprecise definition of a hyperelliptic component of k -differentials is given in Section 4.Roughly speaking, it arises from the locus of k -differentials on hyperelliptic Riemannsurfaces with zeros and poles at some Weierstrass points and hyperelliptic conjugate OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 3 pairs such that the dimension of this locus is equal to the dimension of the ambi-ent stratum. The following result gives a complete classification of such hyperellipticcomponents for all k . Theorem 1.1.
Let µ = (2 m , . . . , m r , l , l , . . . , l s , l s ) be a partition of k (2 g − (withpossibly negative entries). Then the stratum Ω k M g ( µ ) has a hyperelliptic componentif and only if µ is one of the following types: • µ = (2 m , m ) with one of the m i being negative, or m , m > and k ∤ gcd( m , m ) , • µ = (2 m, l, l ) with m or l negative, or m, l > and k ∤ gcd( m, l ) , • µ = ( l , l , l , l ) with some l i < , or l , l > and k ∤ gcd( l , l ) , • µ = ( k (2 g − , • µ = ( k ( g − , k ( g − . Next we generalize the spin parity from the case of k ≤ k . Recall that theparity of an abelian differential ω with singularities of even order only can be defined byusing the mod 2 dimension of the space of sections of the half-canonical divisor div( ω ) / k -differential ( X, ξ ) of signature µ for k ≥
2, we considerthe canonical cover ( b X, b ω ) of ( X, ξ ), that is the minimal cover π : b X → X such thatthe pullback of ξ by π to b X equals the k -th power b ω k of an abelian differential b ω . Ifthe abelian differential b ω has singularities of even order only, then we say that µ is ofparity type and define the parity of ξ by using the parity of b ω . It is known that thisparity can distinguish connected components of the strata for k = 1 (see [KZ03]) butnot for k = 2 (see [Lan04]). Below we show that this dependence on the parity of k holds in general. Theorem 1.2.
Let µ be a signature of parity type for k -differentials with g ≥ . If k is even, then the parity is an invariant of the entire primitive stratum Ω k M g ( µ ) prim .If k is odd, then there exist components of Ω k M g ( µ ) prim with distinct parities, exceptfor the special stratum Ω M (6) prim which is connected. Note that in general the locus of k -differentials with the same parity in Ω k M g ( µ ) prim can possibly be disconnected, as there might be some other structures to further dis-tinguish its components such as the hyperelliptic structure. It would be interestingto know whether the generalized hyperelliptic and parity structures are sufficient toclassify all connected components of the strata. We provide an evidence by using thesestructures to classify connected components of the strata of quadratic differentials withat least one pole of higher order (i.e. quadratic differentials of infinite area). Theorem 1.3.
For genus g ≥ and at least one pole of order ≥ , connected compo-nents of the strata Ω M g ( µ ) prim of primitive quadratic differentials can be described asfollows: (i) If µ is one of the following types: • (2 n, − l ) , • (2 n, − l, − l ) , • ( n, n, − l ) , • ( n, n, − l, − l ) , CHEN AND GENDRON in all of which n and l are not both even, then Ω M g ( µ ) prim has two connectedcomponents, one being hyperelliptic and the other non-hyperelliptic. (ii) For all other µ the primitive stratum Ω M g ( µ ) prim is connected. A more comprehensive statement of the above result including connected componentsarising from squares of abelian differentials can be found in Theorem 7.1 and Table 1.
Applications.
Besides obvious relations with surface dynamics and Teichm¨uller the-ory, connected components of the strata of differentials can be used to study the bi-rational geometry and tautological rings of various moduli spaces (see e.g. [Mul17;Mul18; Bar18; Bae+20]). In particular, Mullane used our classification of connectedcomponents of the strata of meromorphic quadratic differentials to construct extremaland rigid cycles in the moduli space of stable curves with marked points (see [Mul19]).Moreover, Masur–Veech volumes for the strata of abelian and quadratic differentialscan be generalized to all k -differentials of finite area (see [Ngu19]), hence knowing con-nected components of the strata of k -differentials can provide refined information forrelevant volume and intersection calculations (see e.g. [Che+20] for volumes of hyperel-liptic and spin components of abelian differentials). In addition, connected componentsof the strata of k -differentials can provide interesting loci in the strata of abelian differ-entials via the canonical cover, and higher-order differentials (e.g. cubic differentials)often correspond to other geometrically meaningful structures (e.g. real projectivestructures). We thus expect that the results and techniques in this paper can motivatenew discoveries along this circle of ideas. Organization.
This paper is organized as follows. In Section 2 we review the notion ofmulti-scale k -differentials in [CMZ19] and interpret it from our viewpoint. In Section 3we generalize two important constructions in the classification of connected componentsknown for the case k ≤ k ≥
3. In Section 4 we define hyperelliptic componentsand characterize the strata of k -differentials that possess a hyperelliptic component,thus proving Theorem 1.1. In Section 5 we define parity and show that there existcomponents of certain strata of k -differentials which are distinguished by this parityinvariant, thus proving Theorem 1.2. In Section 6 we study adjacency of the strataof quadratic differentials from the viewpoint of merging zeros or poles. In Section 7we study quadratic differentials with arbitrary poles and prove Theorem 1.3 aboutthe classification of connected components of the corresponding strata. Finally in theAppendix we describe an approach to compute the parity of k -differentials in genuszero and one, which motivates a number-theoretic question of independent interest(see Conjecture A.10). Notations.
We identify (compact) Riemann surfaces with smooth (complex algebraic)curves and freely interchange our terminology between them. Let ξ be a k -differentialon a Riemann surface. A singularity of order m ≥ m <
0) is called an analyticzero (resp. analytic pole ) of η . A singularity of order m > − k (resp. m ≤ − k ) iscalled a metric zero (resp. metric pole ) of ξ . A metric zero (resp. metric pole) hasa neighborhood of finite (resp. infinite) area under the flat metric induced by ξ . Forconvenience we also use − m as the order for an analytic pole of order m , e.g., a pole oforder − OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 5 marked point), then its connected components correspond to bijectively those of thestratum by forgetting this ordinary point. Hence we can assume that all entries of asignature are nonzero. Similarly there is a bijection between connected components ofa stratum with singularities labeled or not labeled. For convenience we consider labeledsingularities. We will often use Ω k M g ( n , . . . , n r , − l , . . . , − l s ) to denote the stratumof k -differentials with (analytic) zeros and poles of order n i and l j respectively, and wewill specify in the context when we treat a singularity under the metric sense. Acknowledgements.
We thank Martin M¨oller for helpful discussions on related top-ics. D.C. thanks Dubi Kelmer for interesting conversations on Conjecture A.10. Wealso thank the institutes AIM, CMO and MFO as well as relevant workshop organizersfor inviting and hosting us to work together.2.
Multi-scale and marked k -differentials In this section we review the notion of multi-scale k -differentials together with animportant ingredient called (prong) marked k -differentials.2.1. Multi-scale k -differentials. For the case of abelian differentials ( k = 1) thenotion of multi-scale differentials was introduced in [Bai+19b], and it was extendedto the case of k ≥ k -differentials gives a smooth compactification ofthe strata of k -differentials. For the reader’s convenience, below we review their basicproperties and interpret them from our viewpoint. Definition 2.1. A multi-scale k -differential ( X, z , ξ, , σ ) of type µ consists of(i) a stable pointed curve ( X, z ) with an enhanced level structure on the dualgraph Γ of X ,(ii) a twisted k -differential ( X, z , ξ ) of type µ with a k -prong-matching σ compatiblewith the enhanced level structure.We remark that the idea of twisted differentials (without prong-matching) was knownearlier (see [FP18; Gen18; Che17; Bai+18]). The notion of prong-matching is rela-tively new and plays a key role for the smoothness of the moduli space of multi-scale k -differentials. In particular, the compatibility condition in (ii) requires a k -prong-matching to satisfy a global k -residue condition (see [Bai+19a, Definition 1.4]) so thatthe resulting multi-scale k -differential can be smoothed into the interior of the corre-sponding stratum of k -differentials. For the purpose of our applications, we mainlyfocus on the explanation of k -prong-matching (see [Bai+19b, Section 5.4] and [CMZ19,Section 3.2] for more details).Let ξ be a k -differential on a Riemann surface X which locally near a point p is putin standard form(2.1) φ ∗ ( ξ ) = z κ (cid:0) dzz (cid:1) k if κ > k ∤ κ , (cid:0) sdzz (cid:1) k if κ = 0, (cid:0) z κ/k + s (cid:1) k (cid:0) dzz (cid:1) k if κ < k | κ CHEN AND GENDRON where s ∈ C (and s = 0 in the case κ = 0). In particular, ξ has a zero or pole oforder κ − k at p . The k -residue of ξ at p is defined as Res kp ξ = s k in the last twocases and zero in the first case. In the case κ = 0, we define the (incoming) k -prongs by the | κ | tangent vectors e π i j/ | κ | ∂∂z and the (outgoing) k -prongs by − e π i j/ | κ | ∂∂z for j = 0 , . . . , | κ | −
1. At a pole of order k (i.e. κ = 0), we do not need to define k -prongs.Note that in the case of κ < k | κ , the choice of a k -prong σ gives a consistentway to choose a k -th root of the k -residue. More precisely, we define the k -th rootinduced by σ as the k -th root s of Res kp ξ such that the k -prong σ is horizontal underthe flat metric induced by the k -th root (cid:0) z κ/k + s (cid:1) dzz of the standard form of ξ .There exists a canonical cover π : b X → X of degree k such that π ∗ ξ = b ω k for anabelian differential b ω on b X . For any primitive k -th root of unity ζ , there is a decktransformation τ : b X → b X such that τ ∗ b ω = ζ b ω and the map π is given by taking thequotient of b X by the group action generated by τ . Consider a singularity p of ξ whichhas order = − k (i.e. κ = 0). For a k -prong σ of ξ at p , define the pullback of σ as theset of k equivariant tangent vectors at π − ( p ) which project to σ . In particular, thereis exactly one vector in the pullback for each direction 2 jπ/k for j = 0 , . . . , k − k, κ ) preimages of p and at each preimagethere are k/ gcd( κ, k ) preimages of σ in the pullback.Given a vertical edge e of the enhanced level graph Γ, define a (local) k -prong-matching σ e to be a cyclic order-reversing bijection between the k -prongs at the upperand lower ends of e . A (global) k -prong-matching is a collection σ = ( σ e ) e ∈ E (Γ) v of local k -prong-matchings at every vertical edge.Next we define a prong-matched twisted differential ( b X, b ω, b σ ) associated to a prong-matched twisted k -differential ( X, ξ, σ ) via the cover π : b X → X . Let ( X v , ξ v ) be therestriction of ξ to each irreducible component X v represented by a vertex v of the dualgraph, with the canonical cover ( b X v , b ω v ). We want to glue the preimages of the nodesof X to form ( b X, b ω ) and then define the prong-matching b σ for b X . For each horizontalnode of X , we glue its preimages in a way such that the sum of the residues of b ω at the two branches of each preimage node is equal to zero. For each vertical nodeof X , we glue its preimages that have the pullback of σ in the same directions. Theprong-matching b σ of b X is then given by identifying the prongs in the same direction. Inother words (as in [CMZ19, Section 3.2]), b σ is a prong-matching for the twisted abeliandifferential b ω which is equivariant with respect to the action τ and consistent with the k -prong-matching σ via the cover π . In particular, b σ and σ determine each other.We summarize the above discussion in the following notion. Definition 2.2.
Let ( X, z , ξ , σ ) be a multi-scale k -differential of type µ on X . Thenthe (associated) canonical cover is the multi-scale differential ( b X, b z , b ω, b , b σ ) togetherwith the map π : b X → X such that ( b X, b ω, b σ ) and π : b X → X is defined as in the previousparagraph, b z is the preimage of z under π and b is the order such that b X i b b X j if andonly if X i X j .Note that a prong-matching σ of X gives a welded surface X σ by using σ to identifyisometrically the boundary circles out of the real oriented blowup at each vertical nodeof X (as explained in [Bai+19b, Section 5.3]). We say that X σ is of abelian type if the OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 7 canonical cover b X consists of k connected components, which means ( b X, b ω, b σ ) is the k -th power of a multi-scale (abelian) differential.There is a natural action on multi-scale k -differentials that do not modify the as-sociated welded surfaces. Choose a level, and multiply all differentials at that levelby a nonzero complex number c as well as rotate all prongs of these differentials bythe argument of c (see [Bai+19b, Section 6.1] for a detailed discussion). This inducesnaturally an equivalence relation on the set of k -prong-matchings of a multi-scale k -differential, and two k -prong-matchings are equivalent if and only if they are in thesame orbit of the action. In the following we will use σ to denote an equivalence classof k -prong-matchings .There is a crucial global k -residue condition (see [Bai+19a, Definition 1.4]) thatjustifies when a k -prong-matched twisted differential is compatible with the enhancedlevel graph (i.e. when a k -prong-matched twisted differential can be smoothed intothe interior of the corresponding stratum). Below we reformulate the global k -residuecondition from our viewpoint. Proposition 2.3.
Let ( X, z , ξ , σ ) be a multi-scale k -differential of type µ on X .The k -prong-matching σ is compatible with the enhanced level graph if and only if itsatisfies the global k -residue condition . Namely, for every level L and every connectedcomponent Y of Γ >L , one of the following conditions holds: (i) Y contains a marked pole of ξ . (ii) Y contains a vertex v such that ξ v is not a k -th power of an abelian differential. (iii) (“Horizontal criss-cross in Y ”) For every vertex v of Y the k -differential ξ v isthe k -th power of an abelian differential ω v . Moreover, for every choice of acollection of k -th roots of unity { ζ v : v ∈ Y } there exists a horizontal edge e in Y where the differentials { ζ v ω v } v ∈ Y do not satisfy the matching residue condition. (iv) (“Compatibility of the k -prong-matching”) For every Y such that the weldedsurface Y σ is of abelian type, the k -residues at the edges e , . . . , e N joining Y to Γ = L satisfy the equation (2.2) N X i =1 s i = 0 where s i is the k -th root of Res kq − ei ξ v − ( e i ) induced by the k -prong-matching σ . Note that the left-hand side of Equation (2.2) is a factor of the polynomial P n,k defined in [Bai+19a, Equation (1.1)]. Moreover, our items (i)–(iii) are identical to thecorresponding items of [Bai+19a, Definition 1.4], while our item (iv) combines bothitems (iv) and (v) therein, rephrased in terms of k -prong-matching.2.2. Marked k -differentials and k -prong-matchings. We will show that in somecases there is a unique equivalence class of k -prong-matchings. To do this, we firststudy k -differentials together with some choices of prongs.A k -differential with a choice of k -prongs at some singularities is called a (partially)marked k -differential , which is an important ingredient in the notion of multi-scale k -differentials. In this section we generalize some results of [Boi20] in the classificationof connected components of the strata of (partially) marked k -differentials. CHEN AND GENDRON
Given a stratum of k -differentials, we first consider the case when a unique singularity(of order = − k ) is marked with a k -prong. Lemma 2.4.
Let C be a connected component of a stratum of k -differentials. Thenthe component C marked parameterizing k -differentials in C marked with a k -prong at auniquely chosen singularity is connected.Proof. Since C is connected, it suffices to show that we can join any two fiber pointsin C marked over a k -differential ( X, ξ ) via a continuous path in C . This can be doneby using the continuous family of marked k -differentials ( X, e i t ξ, e − i t/κ ( v )) for t ∈ R ,where κ − k is the order of the singularity, v is the marked k -prong and e − i t/κ ( v ) meansturning the tangent vector v in the reverse direction by the angle t/κ under the flatmetric. In particular for t = 2 π , the k -differential turns back but the k -prong turns tothe next one. (cid:3) Next we consider the case when two singularities of relatively prime orders are markedwith prongs.
Lemma 2.5.
Let C be a connected component of a stratum of k -differentials. Thenthe component C marked parameterizing k -differentials in C marked with prongs at twochosen singularities whose orders plus k are relatively prime is connected. Note that the relatively prime assumption is necessary in Lemma 2.5 (see [Gen18,Corollary 7.9]).
Proof.
The proof is similar to the one of Lemma 2.4, taking into account that theelement (1 ,
1) generates the group Z /κ × Z /κ for κ and κ relatively prime, where κ i − k is the order of each singularity. (cid:3) From the proofs of Lemmas 2.4 and 2.5 we can deduce two useful consequences asfollows.
Corollary 2.6.
Given a twisted k -differential with two levels and a unique edge betweenthem, all k -prong-matchings are equivalent on this twisted k -differential. Corollary 2.7.
Given a twisted k -differential with two levels and two edges of prongnumbers κ and κ between them, if gcd( κ , κ ) = 1 , then all k -prong-matchings areequivalent on this twisted k -differential. Basic operations on k -differentials In this section we generalize two operations, called breaking up a (metric) zero and bubbling a handle , originally due to Kontsevich–Zorich [KZ03] for holomorphic differ-entials and further studied by Lanneau [Lan08] for quadratic differentials with metriczeros and by Boissy [Boi15] for meromorphic differentials. To do this we mix theviewpoints of flat geometry with algebraic geometry, as these operations correspond tosmoothing certain multi-scale k -differentials reviewed in Section 2. Then we study indetail the properties of these operations. OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 9 Breaking up a metric zero.
Recall that a metric zero of a k -differential hassingularity order > − k . We would like to break up a metric zero of order n into r distinct metric zeros of order n , . . . , n r where n = n + · · · + n r .Let ( X , ξ ) be a k -differential with a metric zero z of order n . Take another k -differential ( P , ξ ) in the stratum Ω k M ( n , . . . , n r , − n − k ). Identifying z with thepole p of order − n − k of ξ , we obtain a twisted k -differential ( X, ξ ), as illustratedin Figure 1. The multi-scale k -differential is obtained by taking the unique equivalenceclass of k -prong-matchings σ at the node (as shown in Corollary 2.6). Then we definethe operation of breaking up the metric zero z as the smoothing of the multi-scale k -differential ( X, ξ, σ ) into the respective stratum of k -differentials. n n r ... X P n − n − k Figure 1.
The multi-scale k -differential used in the operation of break-ing up a metric zero.By the global k -residue condition ( X, ξ, σ ) is always smoothable except for one case.This exceptional case occurs when ξ is the k -th power of a holomorphic (abelian)differential and the k -residue of ξ at p is nonzero. By [GT20, Th´eor`eme 1.10] thestratum Ω k M ( n , . . . , n r , − n − k ) does not contain any k -differential ξ with zero k -residue at p if and only if it is of the type Ω k M ( n , n , − n − k ) with k | n but k ∤ n i for i = 1 ,
2. Therefore, we obtain the following conclusion.
Proposition 3.1.
The only non-realizable case of breaking up a metric zero is forbreaking up a zero of the k -th power of a holomorphic differential into two zeros oforder not divisible by k . Bubbling a handle.
This operation allows us to increase the genus of a k -differential by one. It adds 2 k to the order of a metric zero and keep the other singularityorders unchanged.Let ( X , ξ ) be a k -differential in Ω k M g ( n , . . . , n r , − l , . . . , − l s ), where z is themetric zero of order n . Let X be an irreducible rational one-nodal curve with a k -differential ξ having a metric pole p of order − n − k , a metric zero z of order n + 2 k , and two poles of order − k at the two branches N and N of the node such thatthe k -residues R and R of ξ at N and N satisfy the matching residue condition(3.1) R = ( − k R . Identifying the pole p with the zero z and putting the unique k -prong-matchingequivalence class σ at the resulting node, we obtain a multi-scale k -differential ( X, ξ, σ )which is illustrated in Figure 2. The operation of bubbling a handle at the metric zero zz p N ∼ N X X Figure 2.
The multi-scale k -differential used in the operation of bub-bling a handle. z is the smoothing of the multi-scale k -differential ( X, ξ, σ ) into the stratum of genus g + 1 k -differentials Ω k M g +1 ( n + 2 k, n , . . . , n r , − l , . . . , − l s ).By the global k -residue condition ( X, ξ, σ ) is always smoothable except for the casewhen ξ is the k -th power of a holomorphic (abelian) differential and the k -residue of ξ at p is nonzero. Therefore, we obtain the following conclusion. Proposition 3.2.
If the underlying k -differential is not the k -th power of a holomorphicdifferential, then bubbling a handle at a metric zero is always realizable. Let us explain the exceptional case, which will be important in Section 3.3. If ξ isthe k -th power of a holomorphic differential, then n is divisible by k . Consequently theorder of every singularity of ξ is divisible by k . This implies that ξ is the k -th power ofa meromorphic differential ω on X . By the residue theorem, the k -residue of ξ = ω k at p is zero if and only if the residues of ω at N and N are opposite (i.e. ω is astable differential of the nodal curve X ). However, this condition is not automaticallyimplied by the matching residue condition of Equation (3.1) (as r k = ( − r ) k does notimply that r + r = 0 for k > X , ω ), which will be discussed in detail inProposition 3.7.3.3. Local properties of the operations.
We start with an observation for bothoperations.
Lemma 3.3.
Let C be a connected component of Ω k M g ( n , . . . , n r , − l , . . . , − l s ) witha metric zero z of order n . Then the connected component C ′ which contains a k -differential obtained by breaking up z or by bubbling a handle at z depends only onthe choice of ( X , ξ ) in the operation. In other words, this lemma says that given the connected component containing( X , ξ ) and given ( X , ξ ), the resulting connected components out of the two opera-tions do not depend on any other choices in the smoothing process. We remark thatthe cases of k = 1 and k = 2 were known in [KZ03] and [Lan08] respectively. Proof.
According to [CMZ19], the moduli space of multi-scale k -differentials is a smoothcompactification of the corresponding stratum. Hence two connected components ofthe stratum cannot contain respectively two boundary points which can be joined by acontinuous path in the boundary, for otherwise the total space would be singular. (cid:3) OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 11 Recall for breaking up a metric zero, if the underlying k -differential ( X , ξ ) is not the k -th power of a holomorphic differential, then there is no k -residue constraint imposedon ( X , ξ ), hence we can continuously vary the choice of ( X , ξ ) in the connectedstratum Ω k M ( n , . . . , n r , − n − k ). Combining with Lemma 3.3 and using the sameproof, we thus obtain the following result. Lemma 3.4.
Suppose we break up a metric zero whose underlying k -differential ( X , ξ ) is not the k -th power of a holomorphic differential. Then the connected component afterthis operation depends only on the connected component containing ( X , ξ ) and doesnot depend on the choice of ( X , ξ ) . On the contrary, we will discuss how the choice of ξ affects the connected compo-nent C ′ obtained after bubbling a handle. In this operation the (projectivized) stra-tum P Ω k M ( n + 2 k, − k, − k, − n − k ) containing ξ (after normalizing X as P ) isone-dimensional. Hence the matching k -residue condition R = ( − k R at the twopoles N and N of order − k determines finitely many choices of ξ up to scale. Forsuch a k -differential ( X , ξ ), the homology group H ( X \ { p , N , N } , z ) is gener-ated by two simple closed curves γ and γ turning around N and N respectively.Without loss of generality we can assume that γ and γ are “horizontal” (self) saddleconnections of the unique metric zero z (where being “horizontal” is up to a k -th rootof unity). They give a partition of the cone angle ( n + 3 k ) πk at z into four angularsectors of respective angle π , s πk , π and s πk , where s and s satisfy that(3.2) 1 ≤ s i ≤ n + 2 k − s + s = n + 2 k. An example of this partition is illustrated in Figure 3. Conversely given such parame-ters s and s , one can recover ξ by using broken half-planes as basic domains for theconstruction of meromorphic differentials as in [Boi15] (and using broken k -planes asbasic domains for general k , see [Bai+19a, Section 2.3]). Definition 3.5.
If ( X , ξ ) belongs to a connected component C , we denote by C ⊕ s the connected component that contains the differentials obtained by bubbling a handleusing the differential ( X , ξ ) with invariant s as above.This notation ⊕ was originally introduced in [Lan08] for quadratic differentials. Remark 3.6. If d | n and d | k , then the operation ⊕ dℓ at a metric zero of order n of a k -differential ξ = η d is the same as the operation ⊕ ℓ at the corresponding metriczero of order n d of the kd -differential η . This is due to ξ = η d in the respective strataof genus zero used in the operations with s = dℓ for ξ and s = ℓ for η .This remark together with Proposition 3.2 and the discussion after it leads to thefollowing characterization of realizable cases of bubbling a handle. Proposition 3.7. If ξ is not the k -th power of a holomorphic differential, then allthe operations ⊕ s for s ∈ { , . . . , n + 2 k − } are realizable. If ξ is the k -th powerof a holomorphic differential, then the realizable cases of bubbling a handle at a zero oforder n = km are the operations ⊕ kℓ for ℓ ∈ { , . . . , m + 1 } .
21 435435 21 21 4356 3465
Figure 3.
Two differentials in Ω M (4 , − , − , −
4) with zero residueat the pole of order −
4, where k = 1, n = 2, and γ and γ are labelledby 1 and 2 respectively. The invariants ( s , s ) are (1 ,
3) or (3 ,
1) for thedifferential on the left and (2 ,
2) for the differential on the right.A technical tool that we will use to bound the number of connected components ofthe strata is the following generalization of [Lan08, Proposition 2.9] (see also [Boi15,Lemma 3.2 and Remark 3.3]).
Proposition 3.8.
Let C be a connected component of Ω k M g ( n, n , . . . , n r , − l , . . . , − l s ) with a metric zero of order n . Then the following statements hold: (i) If ≤ s ≤ n + 2 k − , then C ⊕ s = C ⊕ ( n + 2 k − s ) . (ii) If ≤ s , s ≤ n + 2 k − and s + s < n + 3 k , then C ⊕ s ⊕ s = C ⊕ s ⊕ s . (iii) If ≤ s ≤ n + k − and k + 1 ≤ s ≤ n + 2 k − , then C ⊕ s ⊕ s = C ⊕ ( s − k ) ⊕ ( s + k ) . (iv) If ≤ s ≤ n + 2 k − , ≤ s ≤ n + 4 k − and s − s ≥ k , then C ⊕ s ⊕ s = C ⊕ ( s − k ) ⊕ s . Proof.
The claim (i) is clear due to the symmetry of s and n + 2 k − s in the definitionof the operation ⊕ . In particular, this implies that it suffices to consider the range1 ≤ s ≤ [ n +2 k ] in order to obtain all connected components of the type C ⊕ s .For the remaining claims, we first explain the idea of the proof from the viewpointof algebraic geometry. Let ( X , ξ ) be the k -differential that we want to bubble at themetric zero z of order n . We construct a multi-scale k -differential ( X, ξ, σ ) by gluing atwisted k -differential ( X , ξ ) to ( X , ξ ) at z . The curve X has geometric genus zeroand has two non-separating nodes N i ∼ N ′ i for i = 1 ,
2. The twisted k -differential ξ has poles of order k at the two nodes, a pole at the point glued to X and a zero in OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 13 the smooth locus, with invariant s i for the (horizontal) saddle connection configurationenclosing the pole N i ∼ N ′ i for i = 1 ,
2. The k -prong-matching σ is the unique oneup to equivalence at the separating node joining X and X . The resulting curve X is illustrated on the left of Figure 4. We further degenerate this twisted k -differential( X, ξ ) into two different multi-scale k -differentials as illustrated on the right of Figure 4.These two degenerations (and the inverse smoothing) correspond to respectively thebubbling operation C ⊕ s ⊕ s on one side and the bubbling operation on the otherside of the desired equalities. X zX degeneration zX Figure 4.
Twisted k -differentials underlying the proof of Proposition 3.8.In order to prove (ii), consider the operation C ⊕ s ⊕ s for a multi-scale k -differentialillustrated on the right of Figure 4. As in the previous paragraph, we denote theintermediate lower level twisted k -differential that we obtain by ( X , ξ ) illustrated onthe left of Figure 4. Let γ i and γ ′ i be the saddle connections enclosing the poles N i and N ′ i of X respectively for i = 1 ,
2. After the total operation
C ⊕ s ⊕ s , consider thefollowing list of (outgoing and incoming) rays emanating from the new zero startingfrom γ in cyclic order: γ , . . . , γ ′ , γ ′ , . . . , γ , γ , . . . , γ ′ , γ ′ , . . . , γ , where the angle between γ , γ ′ is s πk , the angle between γ , γ ′ is s πk , and the anglebetween any two adjacent γ i , γ i or γ ′ i , γ ′ i is π . Let a πk and a πk be the remaining anglesbetween γ ′ , γ and between γ ′ , γ , respectively. Then the total angle at the new zero is( s + s + a + a ) πk + 4 π = ( n + 5 k ) πk , which implies that s + s + a + a = n + 3 k .Note that this list of rays can exist if and only if a + a >
0, namely, s + s < n + 3 k .Figure 5 illustrates such an example in the case of k = 2.Under the same cyclic order we can rewrite this list of rays as γ , . . . , γ ′ , γ ′ , . . . , γ , γ , . . . , γ ′ , γ ′ , . . . , γ , which corresponds to the other operation C ⊕ s ⊕ s . Therefore, to verify (ii) it remainsto show that we can shrink γ i and γ ′ i to zero for both i = 1 , γ can cross γ ′ when shrinking γ and γ ′ . Incontrast, by elementary plane geometry if the sum of the angles between γ , γ ′ , between γ ′ , γ and between γ , γ ′ is bigger than 2 π , i.e. if ( a + a + s ) πk > π , then this crossingissue does not occur. Using a + a + s + s = n + 3 k , this inequality is equivalentto s < n + 2 k . Note that s < n + 2 k holds automatically by the assumption on itsrange. This thus verifies (ii). γ ′ γ γ γ ′ γ ′ γ γ ′ γ Figure 5.
A twisted quadratic differential ( X , ξ ) used in the proof ofthe equality C ⊕ ⊕ C ⊕ ⊕ k = 2, n = 2, s = 2, s = 5, and a = a = .1 1 γ γ ′ γ γ ′ Figure 6.
Shrinking γ and γ ′ can make γ and γ ′ cross when s ≥ n + 2 k . The case represented is k = 4, n = − s = 1 and s = 7. Herewe omit the four half-infinite cylinders glued to γ i and γ ′ i .Next we consider (iii) and use the same notation as in the proof of (ii). For theoperation C ⊕ s ⊕ s , consider the following list of rays when going around the zeroof ξ in cyclic order: γ , . . . , γ , γ , . . . , γ ′ , γ ′ , . . . , γ ′ , γ ′ , . . . , γ . By the assumption on the ranges of s and s , such a list can exist. Indeed since k + 1 ≤ s , the saddle connections γ and γ ′ are outside of the infinite annulus cut outby γ and γ ′ (as a neighborhood of the pole N ∼ N ′ of order − k ), and vice versa.Note that the sector corresponding to( γ , . . . , γ , γ , . . . , γ ′ )has angle s πk + 2 π = ( s + k ) πk , where we gain π from the half-infinite cylinderbounded by γ and another π from the special half-disk sectors adjacent to it (in thesense of [EMZ03, Fig. 8]). Moreover, by definition the sector corresponding to( γ , . . . , γ ′ , γ ′ , . . . , γ ′ )has angle s πk . If we first shrink γ and γ ′ to zero, then the angle of this sector becomes s πk − π = ( s − k ) πk , as we lose π for the half-infinite cylinder bounded by γ ′ aswell as another π from the special half-disk sectors adjacent to it. This thus verifiesthe equality claimed in (iii). OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 15 Similarly for (iv), consider the following list of rays going around the zero of ξ incyclic order: γ , . . . , γ , γ , . . . , γ ′ , γ ′ , . . . , γ ′ , γ ′ , . . . γ . Again by assumption such a list can exist. The sector corresponding to ( γ , . . . , γ ′ ) hasangle s πk . Moreover, the sector corresponding to( γ , . . . , γ , γ , . . . , γ ′ , γ ′ , . . . , γ ′ )has angle s πk . If we first shrink γ and γ ′ to zero, then the angle of this sector becomes s πk − π = ( s − k ) πk . This thus verifies the equality claimed in (iv). (cid:3) We now discuss some consequences of Proposition 3.8. For the operation
C ⊕ s ⊕ s ,by using (i) we can restrict to the range 1 ≤ s ≤ (cid:2) n +2 k (cid:3) and 1 ≤ s ≤ (cid:2) n +4 k (cid:3) . Considerfirst the case n >
0. In this case as long as ( s , s ) = ( n +2 k , n +4 k ) (when n is even),we have s + s < n + 3 k . Since (cid:2) n +4 k (cid:3) ≤ n + 2 k − n > s and s by using (ii). In contrast for ( s , s ) = ( n +2 k , n +4 k ), (ii) doesnot apply (as s + s = n + 3 k ), (iv) does not apply (as s − s = k < k ), while (iii)applies with ( s − k, s + k ) = ( s , s ) which does not change the pair. Next considerthe case − k < n ≤
0. If s ≤ n + 2 k −
1, then s + s ≤ (cid:2) n +2 k (cid:3) + n + 2 k − < n + 3 k (as n ≤ s and s by using (ii). In contrastfor n + 2 k ≤ s ≤ (cid:2) n +4 k (cid:3) , it is outside of the allowed ranges in (ii) and (iii). Moreoverfor (iv), s − k ≤ (cid:2) n +4 k (cid:3) − k ≤ n ≤ s , s )for which Proposition 3.8 does not help simplify the related operations. Definition 3.9.
The parameters ( s , s ) in the operation C ⊕ s ⊕ s are called ofbalanced type if ( s , s ) = ( n +2 k , n +4 k ) for n > n + 2 k ≤ s ≤ (cid:2) n +4 k (cid:3) (and 1 ≤ s ≤ (cid:2) n +2 k (cid:3) ) for − k < n ≤ Corollary 3.10.
Let C and C be two connected components with a unique metric zerosatisfying that C = C ⊕ s ⊕ · · · ⊕ s n . Then there exist ≤ s ′ ≤ · · · ≤ s ′ n such that C = C ⊕ s ′ ⊕ · · · ⊕ s ′ n .Proof. Suppose there are two adjacent s i and s i +1 such that s i > s i +1 . Let n be theorder of the unique metric zero of the differentials in C ⊕ s ⊕ · · · ⊕ s i − . Then byProposition 3.8 (i) we can assume that s i ≤ [ n ] + k , hence s i + s i +1 < s i ≤ n + 2 k ,and then we can exchange s i and s i +1 by Proposition 3.8 (ii). (cid:3) Note that the assumption of a unique zero in the above is not essential. As long aswe keep performing the operations ⊕ for a designated zero, the same argument andconclusion still hold.Since we often focus on primitive k -differentials, the following fact is useful to know. Remark 3.11.
Suppose we break up a metric zero or bubble a handle for a primitive k -differential ( X, ξ ). Then the k -differentials we obtain remain to be primitive. Thisis because a (multi-scale) k -differential is primitive if and only if its canonical cover is connected. If after smoothing we obtain a non-primitive k -differential, then the canon-ical cover of the multi-scale k -differential used in the operations is disconnected, whichcontains a disconnected canonical cover over the component ( X, ξ ), thus contradictingthat (
X, ξ ) is primitive.3.4.
Global properties of the operations.
The classification of connected com-ponents of the strata of meromorphic abelian differentials in genus one was given in[Boi15, Theorem 4.1] (see [CC14, Section 3.2] for another viewpoint). Since the canon-ical bundle of a genus one curve is trivial, the same classification holds for connectedcomponents of the strata of k -differentials in genus one for all k .We first generalize the rotation number description in [Boi15, Section 4.2] to mero-morphic k -differentials on genus one curves for all k . Let ( X, ξ ) be a k -differential in astratum Ω k M ( n , . . . , n r , − l , . . . , − l s ) for X of genus one. Suppose γ : S → X is asimple closed curve such that γ does not contain any singularity of ξ . Define a mapΓ : S → S / (cid:10) exp( π i k ) (cid:11) by t γ ′ ( t ) | γ ′ ( t ) | where a unit tangent vector is taken with respect to the flat metric induced by ξ . Wequotient the target S by (cid:10) exp( π i k ) (cid:11) , as the holonomy of the k -translation structureinduced by ξ is contained in this group. Regarding the target of Γ still as S , thismap Γ thus defines the index Ind( γ ) of γ in the usual way. Let ( a, b ) be a symplecticbasis of H ( X, Z ), and choose ( α, β ) as arc-length representatives of ( a, b ) that avoidthe singularities of ξ . As in [Boi15, Definition 4.2], we define the rotation number of a k -differential ( X, ξ ) byrot(
X, ξ ) = gcd(Ind( α ) , Ind( β ) , n , . . . , n r , l , . . . , l s ) . This invariant allows us to describe geometrically the classification of connectedcomponents of the strata of k -differentials in genus one, thus generalizing the viewpointof [Boi15, Theorem 4.3] from k = 1 to all k . Theorem 3.12.
Let S = Ω k M ( n , . . . , n r , − l , . . . , − l s ) be a stratum of k -differentialson genus one curves. Then the rotation number is an invariant for any connectedcomponent of S .Let d be a positive divisor of gcd( n , . . . , n r , l , . . . , l s ) . Then d can be realized as arotation number for a unique connected component of S , except that d = n does notoccur for the stratum Ω k M ( n, − n ) .Moreover when P ri =1 n i > k , any connected component of S can be realized by theoperations of bubbling a handle and breaking up a zero.Proof. The proof of the first claim that the rotation number is an invariant for anyconnected component of S is the same as in [Boi15]. Indeed it suffices to observe thatwhen crossing a singularity of order n i the index of a simple closed curve changes byadding ± n i .Next we prove the remaining claims about the realization of rotation numbers. For n ≥ k + 1 and 1 ≤ t ≤ n −
1, apply the operation of bubbling a handle ⊕ t to a k -differential in the stratum Ω k M ( n − k, − l , . . . , − l s ) with n = l + · · · + l s . Wethen obtain a k -differential in Ω k M ( n, − l , . . . , − l s ). We can construct a symplectic OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 17 basis for the first homology group of the underlying torus as follows. Take α to bethe core curve of the bubbled cylinder and β to be the curve that goes through thenon-separating node and turns around the newborn metric zero of order n . The in-dices of α and β are respectively equal to 0 and t (or n − t ). Now we break up thezero of order n to r metric zeros of order n , . . . , n r . The indices of α and β changecontinuously, and hence they remain unchanged. By taking t to be any positive divisor d of gcd( n , . . . , n r , l , . . . , l s ) (with d < n ), we thus obtain a k -differential of rotationnumber d in Ω k M ( n , . . . , n r , − l , . . . , − l s ) when n = P ri =1 n i > k . Note that therotation number n never arises in this way.For the case n = k one can check it directly by using Figure 7, where α and β can bechosen as closed paths connecting the middle points of the saddle connections 1 and 2respectively and both have index d for any 1 ≤ d < k and d | k . Similarly by usingFigure 8 one can check it directly for the case 2 ≤ n < k . d πk Figure 7. A k -differential in Ω k M ( k, − k ) of rotation number d . d πk n πk Figure 8. A k -differential in Ω k M ( n, − n ) of rotation number d for2 ≤ n < k .The previous discussion provides at least as many connected components as thenumber given in [Boi15, Theorem 4.1]. Therefore, we conclude that each rotationnumber is realizable by a unique connected component, which moreover can be realizedby the operations of bubbling a handle and breaking up the resulting zero when thetotal zero order n > k . (cid:3) We remark that connected components of the strata in genus one can also be classifiedby another invariant from the algebraic viewpoint (see [CC14, Section 3.2]). To defineit, let (
X, ξ ) be a k -differential in a stratum Ω k M ( µ ) with µ = ( m , . . . , m n ) such that P ni =1 m i = 0. Denote by gcd( µ ) = gcd( m , . . . , m n ). Let d be a positive divisorof gcd( µ ) (except that d = n is not allowed for µ = ( n, − n )). We say that ( X, ξ )has torsion number d , if d is the largest integer such that P ni =1 ( m i /d ) z i ∼ X (i.e. P ni =1 ( m i /d ) z i represents the trivial divisor class). Then there is a one-to-onecorrespondence between the connected components of Ω k M ( µ ) and the loci of ( X, ξ )with fixed torsion numbers.It is natural to ask whether the torsion number coincides with the rotation number.This was indeed confirmed in [Tah18, Section 3.4] for abelian differentials. Herein weadapt the same argument and generalize it to higher k . Proposition 3.13.
The rotation number coincides with the torsion number for k -differentials on genus one curves.Proof. If gcd( µ ) = 1, then the stratum Ω k M ( µ ) is irreducible, and by definition therotation number and the torsion number are both equal to 1 in this case. In general,suppose that ( X, ξ ) ∈ Ω k M ( µ ) has torsion number d . It implies that there exists a k -differential ( X, ξ ′ ) in the stratum Ω k M ( µ/d ) with the same underlying pointed curve X . Moreover, up to scaling we can write ξ ′ = f ( z )( dz ) k and ξ = f ( z ) d ( dz ) k with f alocally meromorphic function. By Cauchy’s argument principle, the indices of α and β relative to the metric induced by ξ is d times the indices of these cycles relative tothe metric induced by ξ ′ . It implies that the rotation number of ( X, ξ ) is d times therotation number of ( X, ξ ′ ), and hence the rotation number of ( X, ξ ) is divisible by itstorsion number. The desired claim then follows from the fact that there is a uniqueconnected component of Ω k M ( µ ) for each d , regardless of d being the rotation numberor the torsion number. (cid:3) As a corollary of Theorem 3.12 we obtain the following useful result, which generalizes[Boi15, Proposition 4.4].
Lemma 3.14 (The gcd-trick) . Let S = Ω k M ( n, − l , . . . , − l s ) be a stratum of genuszero k -differentials with a metric zero of order n . If gcd( s , l , . . . , l s ) = gcd( s , l , . . . , l s ) ,then S ⊕ s = S ⊕ s .Proof. Take two k -differentials out of the two bubbling operations respectively. Take α to be the core curve of the bubbled cylinder and β to be the curve that goes throughthe non-separating node and turns around the newborn metric zero of order n + 2 k .The indices of α and β are respectively equal to 0 and s i (or n + 2 k − s i = P sj =1 l j − s i ).Hence by the gcd assumption both k -differentials after the two bubbling operationshave the same rotation number. The claim thus follows from Theorem 3.12. (cid:3) In the same vein, the following result will be useful for classifying connected compo-nents of the strata.
Lemma 3.15.
Let C be a connected component of Ω k M g ( n, − l , . . . , − l s ) with a metriczero of order n and s ≥ . If gcd( s , n + 2 k ) = gcd( s , n + 2 k ) , then C ⊕ s = C ⊕ s .Proof. In the bubbling operation (see Figure 2), we can first smooth the non-separatingnode of ( X , ξ ) to obtain a genus one k -differential ( X ′ , ξ ′ ) in Ω k M ( n + 2 k, − n − k ).By the gcd assumption, such k -differentials ( X ′ , ξ ′ ) for the operations ⊕ s i for i = 1 , OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 19 have the same rotation number. Hence they belong to the same connected component ofthe stratum Ω k M ( n +2 k, − n, − k ). Moreover, since the top level components ( X , ξ )contain poles, they are not k -th powers of holomorphic differentials. Hence the global k -residue condition is satisfied, which implies that these multi-scale k -differentials aresmoothable. Therefore, the k -differentials obtained by smoothing further the separatingnode between X and X ′ can be connected by a continuous path in the space of multi-scale k -differentials of signature ( n + 2 k, − l , . . . , − l s ). This implies that they belongto the same connected component of the corresponding stratum. (cid:3) Hyperelliptic components of the strata of k -differentials In this section we define and classify hyperelliptic components of the strata of k -differentials. We begin with extending the known definition of hyperelliptic componentsfrom the case of abelian and quadratic differentials to all k -differentials. Definition 4.1. A k -differential ( X, ξ ) is called hyperelliptic if X is a hyperelliptic curveand ξ is ( − k -invariant under the hyperelliptic involution. A connected componentof the strata of k -differentials is called a hyperelliptic component if every k -differential( X, ξ ) in this component is hyperelliptic.In affine coordinates a hyperelliptic curve X of genus g can be represented by theequation x = ( y − y ) · · · ( y − y g +2 ) with y , . . . , y g +2 distinct fixed points. The map( x, y ) y is the hyperelliptic double cover and ( x, y ) ( − x, y ) is the hyperellipticinvolution ι . The points w i = (0 , y i ) for i = 1 , . . . , g + 2 give the 2 g + 2 Weierstrasspoints of X . For a k -differential ξ on X satisfying that ι ∗ ξ = ( − k ξ , if p is a singularityof ξ which is not a Weierstrass point, then clearly the conjugate p ′ = ι ( p ) has to be asingularity of ξ with the same order. If a Weierstrass point is a singularity of order n ,since ι ∗ x n ( dx ) k = ( − n + k x n ( dx ) k , for ξ to be ( − k -invariant we conclude that n mustbe even. Therefore, the associated k -canonical divisor of a hyperelliptic k -differential( X, ξ ) is of the form r X i =1 m i w i + s X j =1 l j ( p j + p ′ j )where p j and p ′ j are hyperelliptic conjugates. Conversely, such a k -canonical divisordetermines a hyperelliptic k -differential up to a scalar multiple.Note that the locus of hyperelliptic k -differentials with a given signature can be alower dimensional subspace in the corresponding stratum. In order to obtain a hyper-elliptic component, we need to check that the dimension of the locus of hyperelliptic k -differentials with a given signature equals the total dimension of the correspondingstratum. Using this strategy we can classify hyperelliptic components of the strata of k -differentials as follows, thus proving Theorem 1.1. Theorem 4.2.
Let µ = (2 m , . . . , m r , l , l , . . . , l s , l s ) be a partition of k (2 g − (withpossibly negative entries). Then the stratum Ω k M g ( µ ) has a hyperelliptic componentif and only if µ is one of the following types: • µ = (2 m , m ) with one of the m i being negative, or m , m > and k ∤ gcd( m , m ) , • µ = (2 m, l, l ) with m or l negative, or m, l > and k ∤ gcd( m, l ) , • µ = ( l , l , l , l ) with some l i < , or l , l > and k ∤ gcd( l , l ) , • µ = ( k (2 g − , • µ = ( k ( g − , k ( g − .Proof. We first assume that g ≥
2. Consider the space Hyp k ( µ ) parameterizing genus g hyperelliptic curves with k -canonical divisors of the form r X i =1 m i w i + s X j =1 l j ( p j + p ′ j )where w i are Weierstrass points and p j , p ′ j are hyperelliptic conjugates. It is easy tosee that Hyp k ( µ ) is irreducible with dimensiondim Hyp k ( µ ) = 2 g − s. Since every connected component of P Ω k M g ( µ ) has dimension ≥ g − r + 2 s (seee.g. [Bai+19a, Theorem 1]), if Hyp k ( µ ) gives a connected component of P Ω k M g ( µ ),then we have 2 g − s ≥ g − r + 2 s. It implies that r + s ≤
2. We now treat each case separately.Consider the case r = 2 and s = 0, i.e., µ = (2 m , m ) where m + m = k ( g − m and m is negative, or if m , m > k ∤ gcd( m , m ), thendim P Ω k M g (2 m , m ) = 2 g − k (2 m , m ) . Hence Hyp k (2 m , m ) gives rise to a connected component. If m , m > k | gcd( m , m ), then P Ω k M g (2 m , m ) has some component of dimension 2 g arisingfrom the k -th powers of abelian differentials of signature (2 m /k, m /k ). In this caseany differential in Hyp k (2 m , m ) is also the k -th power of an abelian differential,because ( m /k )(2 w ) + ( m /k )(2 w ) is a canonical divisor. Since the dimension ofHyp k (2 m , m ) is 2 g − < g , Hyp k (2 m , m ) does not give a component. We remarkthat if (2 m , m ) = ( k ( g − , k ( g − r = 0 and s = 1 to be discussed later.Consider s = 2 and r = 0, i.e., µ = ( l , l , l , l ). If some l i <
0, or if l , l > k ∤ gcd( l , l ), thendim P Ω k M g ( l , l , l , l ) = 2 g + 1 = dim Hyp k ( l , l , l , l ) . Hence the locus Hyp k ( l , l , l , l ) gives rise to a connected component. If l , l > k | gcd( l , l ), then P Ω k M g ( l , l , l , l ) has some component of dimension 2 g + 2arising from the k -th powers of abelian differentials of signature ( l /k, l /k, l /k, l /k ).In this case any differential in Hyp k ( l , l , l , l ) is also the k -th power of an abeliandifferential, because the divisor ( l /k )( p + p ′ ) + ( l /k )( p + p ′ ) is canonical. Bycomparing dimensions it implies that Hyp k ( l , l , l , l ) does not give a component inthis case. OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 21 Consider s = r = 1, i.e., µ = (2 m, l, l ). If m or l is negative, or if m, l > k ∤ gcd( m, l ), thendim P Ω k M g (2 m, l, l ) = 2 g = dim Hyp k (2 m, l, l ) . Hence Hyp k (2 m, l, l ) gives rise to a connected component. If m, l > k | gcd( m, l ),then P Ω k M g (2 m, l, l ) has some component of dimension 2 g + 1 arising from the k -thpowers of abelian differentials of signature (2 m/k, l/k, l/k ). In this case any differentialin Hyp k (2 m, l, l ) is also the k -th power of an abelian differential, because the divisor( m/k )(2 w )+ ( l/k )( p + p ′ ) is canonical. Hence Hyp k (2 m, l, l ) does not give a component.Consider r = 1 and s = 0, i.e., µ = ( k (2 g − k ( k (2 g − k -th power of an abelian differential of signature (2 g − g − w is a canonical divisor. Hence Hyp k ( k (2 g − P Ω M g (2 g − s = 1 and r = 0, i.e., µ = ( k ( g − , k ( g − k ( k ( g − , k ( g − k -th power of an abelian differential of signature( g − , g − g − p + p ′ ) is a canonical divisor. Hence Hyp k ( k ( g − , k ( g − P Ω M g ( g − , g − g = 1, Weierstrass points can be interpreted as fixed points under theelliptic involution (i.e. 2-torsion points on curves of genus one), and in this sense theabove proof goes through without any change. Finally for the case g = 0, a double coverfrom P to P can be determined by specifying two ramification points in the domain,or one ramification point with a pair of conjugate points, or two pairs of conjugatepoints, which correspond to the types of signatures in the claim. (cid:3) Recall the operation ⊕ introduced in Definition 3.5. We conclude this section byanalyzing when this operation gives hyperelliptic components. Lemma 4.3.
Let S be the hyperelliptic component of the stratum Ω k M g − (2 m , m ) or Ω k M g − (2 m , l, l ) listed in Theorem 4.2 such that the singularity of order m is ametric zero. Then the hyperelliptic component of the stratum Ω k M g (2 m + 2 k, m ) or Ω k M g (2 m + 2 k, l, l ) can be given by S ⊕ ( m + k ) .Proof. Let (
X, ξ ) be a hyperelliptic k -differential of genus g obtained by the operation ⊕ s of bubbling a genus one differential ( X , ξ ) at the metric zero of order 2 m of a genus g − X , ξ ) (see Figure 2). The hyperelliptic involution ι actson both ( X i , ξ i ). Take a homology class α in X represented by a closed path that goesthrough the sector of angle s πk and the bubbled cylinder, so that the index of α is s .One can represent − α by another closed path that goes through the complementarysector of angle (2 m + 2 k − s ) πk and the bubbled cylinder, so that the index of − α is2 m + 2 k − s . Since ι ∗ α = − α , we conclude that s = 2 m + 2 k − s , hence s = m + k .For the converse, it suffices to show that a k -differential obtained by applying thebubbling operation ⊕ ( m + k ) to a hyperelliptic k -differential of genus g − g + 2 fixed points. This can be verified for the twisted k -differential( X, ξ ) as the nodal union of ( X i , ξ i ) in the definition of bubbling a handle, and thesmoothing process preserves this property (see [Gen18, Section 6] for more details). In particular, 2 g − X . The remaining three fixed points come from the zeroof ξ , the center of the bubbled cylinder, and the middle point of the saddle connectiontransversal to the boundary of the cylinder. (cid:3) Parity of the strata of k -differentials In this section we define a parity for k -differentials and characterize in Theorem 5.2the strata of k -differentials that have components distinguished by this parity invariant.5.1. The parity.
Recall from [KZ03] (and [Boi15] in the meromorphic case) that fora stratum Ω M g (2 m , . . . , m n ) of abelian differentials with singularities of even orderonly, we can define an invariant in the following way. Given ( X, ω ) in this stratum, the parity of ω is defined as(5.1) Φ( ω ) := h (cid:0) X, div ( ω ) (cid:1) (mod 2) . Alternatively, let ( α , . . . , α g , β , . . . , β g ) be a symplectic basis of H ( X, Z /
2) whichdoes not meet the singularities of ω . Then the parity of ω can also be defined as theparity of the Arf-invariant(5.2) Φ( ω ) := g X i =1 (Ind ω ( α i ) + 1)(Ind ω ( β i ) + 1) (mod 2) . The notion of parity was extended to quadratic differentials in [Lan04, Section 3.2],and we now generalize it to k -differentials for all k . Let Ω k M g ( µ ) be a stratum of k -differentials with µ = ( m , . . . , m s , − l , . . . , − l r ). Given a (primitive) k -differential( X, ξ ) in this stratum, we denote by ( b X, b ω ) the (connected) canonical k -cover of ( X, ξ )(see [Bai+19a]). For any integer m , we define b m = m + k gcd( m,k ) −
1. A singularity of ξ oforder m gives rise to gcd( m, k ) singularities of order b m of b ω . If b m i and d − l j are even forall i and j , then the parity Φ( ξ ) of ( X, ξ ) is defined to be the parity of the canonicalcover ( b X, b ω ).The following result describes when the canonical cover of a k -differential has sin-gularities of even order only. The proof is an easy computation left to the reader. Weuse the 2-adic valuation of n as the highest exponent v ( n ) such that 2 v ( n ) divides n . Proposition 5.1.
Let ( X, ξ ) be a k -differential in the stratum Ω k M g ( µ ) . Then thecanonical cover of ( X, ξ ) has only even order singularities if and only if the -adicvaluation of every entry of µ is not equal to v ( k ) . The strata satisfying the hypotheses of Proposition 5.1 are called of parity type . Wenow state the main result of this section, which combining with Theorem 5.10 aboutthe special strata in genus two refines Theorem 1.2 in the introduction.
Theorem 5.2.
Let S = Ω k M g ( µ ) prim be a stratum of primitive k -differentials of paritytype with g ≥ . If k is even or S = Ω M (6) prim , then the parity is an invariant ofthe entire stratum S . If k is odd and S = Ω M (6) prim , then there exist componentsof S with distinct parity invariants. OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 23 Note that in Theorem 5.2 we do not claim that the parity is enough to distinguishall connected components of the strata of parity type, as there might be some otherinvariants such as hyperelliptic structures. In other words, the locus with the sameparity invariant in a stratum of parity type may still be disconnected.We separate the proof of Theorem 5.2 in three steps. In Section 5.2 we treat thecase of even k and in Section 5.3 the case of odd k , except several sporadic strata ingenus two which are treated in Section 5.4.Before starting the proof of Theorem 5.2, we make two remarks. The first onegeneralizes the definition of parity to non-primitive k -differentials. Remark 5.3.
For our applications it will be useful to consider parities for non-primitive k -differentials as well. Suppose ξ = η d where η is a primitive ( k/d )-differential ofparity type. Then we define Φ( ξ ) := d Φ( η ). In this case the canonical cover b X of ξ consists of d connected components, each of which is a (connected) canonical cover of η .Therefore, we can still apply Equations (5.1) and (5.2) to compute Φ( ξ ), replacing X by the disconnected cover b X . Note that if k is odd (and hence d is odd), the definitionreduces to Φ( ξ ) = Φ( η ).The next remark is related to a construction of Boissy. Remark 5.4.
In [Boi15, Section 5.3], Boissy defines a parity for meromorphic abeliandifferentials in the strata Ω M g (2 m , . . . , m n , − , −
1) with m i ≥ i . For suchdifferentials ω , by the residue theorem ω has opposite residues at the two simple poles.Hence one can construct a stable differential by identifying the two poles as a node.The parity of ω is then defined as the parity of (the smoothing of) this stable differentialof (arithmetic) genus g + 1, which can have distinct parities in general.One can try to adapt this construction to the strata of k -differentials of analogoustypes. However, some direct generalizations do not work. For instance, consider thestrata Ω k M g ( m , . . . , m n , − k, − k ) with two poles of order k . Identify the two polesas before. Note that for k >
1, there is no k -residue theorem. Hence the resultingnodal k -differential may not satisfy the matching k -residue condition at the node, andconsequently it may fail to be a smoothable stable k -differential.Alternatively, consider the strata of type Ω k M g ( m , . . . , m n , − k, . . . , − k ) with k even and v ( m i ) = v ( k ) for all i . Given such a k -differential ( X, ξ ), let ( b X, b ω ) bethe canonical cover of ( X, ξ ). We can form a stable differential by identifying pairwisethe preimages of the poles of order k in ( b X, b ω ) that have opposite residues (since k iseven). Then the parity of ( X, ξ ) can be defined as the parity of (the smoothing of)this stable differential. Note that the deck transformation τ of the canonical coverextends to the (equivariant) smoothing, hence the quotient is a k -differential in thestratum Ω k M g ( m , . . . , m n , − k/ , . . . , − k/
2) of parity type, where each pole of order k in ξ yields a pair of poles of order k/ k (i.e. smoothing a simple polar node of the abelian differential b ω in the cover). From the algebraic viewpoint, splitting a pole of order − k into twopoles of order − k/ k -differential of signature( − k, − k/ , − k/
2) with matching k -residue and then smoothing the resulting multi-scale k -differential. By Theorem 5.2, nevertheless, there is a unique parity of k -differentialsin Ω k M g ( m , . . . , m n , − k/ , . . . , − k/
2) for k even. Hence this construction does notprovide distinct parities either.5.2. The case of even k . In this section we study the strata of k -differentials of paritytype when k is even. Proposition 5.5.
Let Ω k M g ( m , . . . , m n ) prim be a primitive stratum of k -differentialswith v ( k ) = v ( m i ) for all i . If k is even, then the parity is an invariant of the stratum. We remark that it was known in [Lan04] that the parity is an invariant for anyprimitive stratum of quadratic differentials of parity type. Moreover, although it wasstated for the case of quadratic differentials with metric zeros only, the same argumenttherein works for meromorphic quadratic differentials with arbitrary poles.
Proof.
Given a k -differential ( X, ξ ), recall that the canonical cover π k : b X → X as-sociated to ξ is the k -cyclic cover of X obtained by taking a k -th root b ω of ξ (seee.g. [Bai+19a, Section 2.1]). Suppose k = dk ′ with 1 < d, k ′ < k . Then we can simi-larly construct an intermediate canonical d -cyclic cover π d : Y → X by taking a d -throot of ξ , that is, π ∗ d ξ = η d for a k ′ -differential η on Y . We can further take the canonical k ′ -cyclic cover π k ′ : b Y → Y such that π ∗ k ′ η = b η k ′ for an abelian differential b η on b Y . Bythe universal property of canonical covers, we have the following commutative diagram(5.3) ( b Y , b η ) ( b X, b ω )( Y, η ) (
X, ξ ) φπ k ′ π d π k where φ : b Y → b X is an isomorphism such that φ ∗ ( b ω ) = ζ b η with ζ a k -th root of unity.Now for even k = 2 d , consider any k -differential ( X, ξ ) in a primitive stratum S of parity type. In the above setting we can take the canonical d -cover ( Y, η ) where η is a primitive quadratic differential of parity type. Note that the parity of ( Y, η ) isan invariant of the respective stratum of quadratic differentials according to [Lan04].It then follows from the commutative diagram (5.3) that the parity of (
X, ξ ) is aninvariant of the stratum S , since the parity of ξ equals the parity of η (both equal tothe parity of b η by definition). (cid:3) Remark 5.6.
For a primitive stratum Ω M g ( m , . . . , m n ) prim of quadratic differentialsof parity type, the parity can be computed by [Lan04, Theorem 4.2] as n + − n − n + is the number of m i ≡ n − is the number of m i ≡ X, ξ ) in a primitive stratum Ω k M g ( m , . . . , m n ) prim of k -differentials ofparity type for even k = 2 d . Note that a singularity of order m in ξ has r = gcd( m, d ) OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 25 preimages under π d , each of order m + kr − η . Combiningwith the above formula it thus gives the parity of the stratum Ω k M g ( m , . . . , m n ) prim .5.3. The case of odd k . In this section we construct k -differentials of both paritiesfor k odd and all singularity orders even, except the strata Ω M (6), Ω M (4 ,
2) andΩ M (2 , ,
2) which will be treated in Section 5.4. The construction goes as follows.We first find some explicit examples in the minimal strata. Then we extend to all stratausing certain multi-scale k -differentials built on these examples. In order to computethe parity of the k -differentials obtained by smoothing the multi-scale k -differentials,we first prove the following result. Lemma 5.7.
For i = 0 , let ( X i , ξ i ) be two k -differentials of parity type such that ξ i = η d i i with η i a primitive ( k/d i ) -differential. Denote by ( X, ξ ) a smoothing of themulti-scale k -differential ( X ′ , ξ ′ , σ ′ ) obtained by gluing a singularity of ξ to a singularityof ξ . Then ( X, ξ ) is of parity type and (5.4) Φ( ξ ) = d Φ( η ) + d Φ( η ) . Proof.
Let b X ′ be the canonical cover of the multi-scale k -differential ξ ′ as defined inDefinition 2.2 and let ( b X, b ω ) be the canonical cover of the smoothing ( X, ξ ) of ( b X ′ , ξ ′ ).Note that b X degenerates to b X ′ by shrinking the vanishing cycles that become thenodes of b X ′ . The k -differential ξ is of parity type since the orders of singularities in thesmooth locus of the multi-scale k -differential are preserved in the smoothing process.Next we construct a symplectic basis of H ( b X, Z / b X i be the canonical coverof ( X i , η i ) for i = 0 ,
1. Then b X ′ contains d copies of b X and d copies of b X . Wecan take a symplectic basis of each copy of H ( b X i , Z /
2) for i = 0 , H ′ of H ( b X, Z / H ( b X, Z / b X together with theirdual cycles. More precisely, let Γ be the dual graph of b X ′ . Then by [ACG11, (9.24)]we know that H ( b X, Z / ∼ = H ′ ⊕ H (Γ , Z / ⊕ H (Γ , Z / H (Γ , Z /
2) canbe identified with the group generated by the vanishing cycles which is dual to thegroup of loops in Γ under the intersection paring. Moreover, it is easy to see that theintersection numbers are zero for any two (not necessarily distinct) vanishing cycles,for any two (not necessarily distinct) loops in Γ, and for any vanishing cycles or anyloops with any elements in H ′ . Therefore, we can take a basis α i of H (Γ , Z /
2) andtheir dual cycles β i as a basis of H (Γ , Z /
2) to complete the desired symplectic basisof H ( b X, Z / b X, b ω ) by using the above symplectic basis of H ( b X, Z /
2) and the Arf-invariant in the definition of parity. From the part of H ′ thecontribution to the parity is already d Φ( η ) + d Φ( η ) as claimed in Equation (5.4).Since the operation of plumbing a multi-scale k -differential does not change the indicesof any closed paths away from a neighborhood of the nodes, it suffices to show that thecycles α i and β i constructed above do not contribute to the parity. Denote by m theorder of the singularity q of ξ at the node joining X and X . Note that the indices ofthe α i cycles are all equal to b m + 1, where b m is the singularity order of each preimageof q in b X . Since by assumption ξ and ξ are of parity type, it implies that b m is even, and hence the index of each α i is odd. It follows that (Ind b ω ( α i ) + 1)(Ind b ω ( β i ) + 1) ≡ (cid:3) We now construct primitive k -differentials with distinct parities in the minimal strataΩ k M g ( k (2 g − prim for k odd, except the stratum Ω M (6) prim . The key tool for theconstruction is the following result. Lemma 5.8.
For k ≥ odd and n ≥ , there exist k -differentials in the primitivestratum Ω k M (2 kn, − kn ) prim with zero k -residue at the pole. Moreover for k ≥ odd, there exist such differentials with both parities.Proof. We first exhibit a primitive k -differential ( X , ξ ) in Ω k M (2 k, − k ) prim withzero k -residue at the pole in Figure 9. Take a symplectic basis ( α, β ) of H ( X , Z /
2) as ( k − a i ) πk β α Figure 9.
A primitive k -differential with zero k -residue at the polein Ω k M (2 k, − k ) prim for a i ∈ (cid:8) , . . . , ⌊ k ⌋ (cid:9) relatively prime to k . Werotate the edges 1 and 2 on the left clockwise by angle 2 a i π/k and thenidentify them respectively with the edges 1 and 2 on the right via trans-lation.in the figure. By elementary geometry the total angle variation of the tangent vectorsto α is equal to 2 π − a i k π , hence the index of α is k − a i by using the definition inSection 3.4. Similarly the index of β is also equal to k − a i . Moreover, since the edgeidentifications are made by rotation of angle 2 a i π/k , the corresponding k -differentialis primitive if and only if a i is relatively prime to k .Let ( b α, b β ) be the preimages of ( α, β ) in the canonical cover ( b X , b ω ). Since the index k − a i of α is relatively prime to k , the preimage b α is connected by gluing k copies of α consecutively. The total angle variation of the tangent vectors to b α is thus k times thatof α , hence the index of b α (with respect to the abelian differential b ω ) is k − a i whichis equal to the index of α . Similarly the index of b β is also k − a i , equal to the indexof β . If k is odd, ( b α, b β ) form a symplectic basis of H ( b X , Z /
2) as they intersect at k points, and hence in this case the parity of the k -differential ξ given in Figure 9 isequal to the parity of a i . This concludes the proof for the case n = 1, since for k = 3we can choose a i = 1 and for odd k ≥ a i to be 1 or 2.To prove for general n , we can modify the above construction as follows. In Figure 9,take a vertical half-infinite ray starting from the middle bullet point and going up, and OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 27 cut the plane open along this ray. Take n − k -differential belongs to Ω k M (2 kn, − kn ).Moreover, the cycles α , β , and their indices are unchanged in this process. Hence theclaim follows from the same analysis as in the previous paragraph. (cid:3) Remark 5.9.
Note that in the above proof if α and β have indices k − a i with a i and k relatively prime, then the rotation number is gcd(2 kn, k − a i ). Hence in this casethe component of Ω k M (2 kn, − kn ) with rotation number gcd(2 kn, k − a i ) has parityequal to the parity of a i . In particular, this confirms a special case of Theorem A.17in the Appendix, where we will study the parities of general strata of k -differentials ingenus zero and one.We now apply Lemma 5.7 to construct k -differentials in Ω k M g ( k (2 g − prim withdistinct parities for the following cases:(i) k ≥ g ≥ k ≥ g ≥ M (12) prim .We first explain the idea of the construction. As in the notation of Lemma 5.7, we take a k -differential ( X , ξ ) in the stratum Ω k M g − ( k (2 g − k -differential ( X , ξ ) inthe stratum Ω k M ( k (2 g − , − k (2 g − prim . Then we form a multi-scale k -differentialby gluing the zero of ξ to the pole of ξ . Choosing ξ and ξ carefully will lead to k -differentials with distinct parities after smoothing the multi-scale k -differentials.In case (i), we choose ξ to be the k -th power of an abelian differential in the stratumΩ M g − (2 g − g ≥
4, there exist such differentials ξ and ξ with distinct parities(see [KZ03]). The construction then follows from Equation (5.4) by gluing to both ξ and ξ the same primitive k -differential ξ in Ω k M ( k (2 g − , − k (2 g − prim withzero k -residue at the pole, which exists by Lemma 5.8. Since ξ is primitive, aftersmoothing the multi-scale k -differentials we thus obtain two primitive k -differentials ofdistinct parities.In case (ii), we choose any ω ∈ Ω M g − (2 g −
4) and take as before ξ = ω k . ByEquation (5.4), we can obtain two distinct parities for ξ by taking two ξ with distinctparities. This is possible for every k ≥ M (6) prim and forma multi-scale 3-differential by gluing to it the third power of an abelian differential( X , ω ) in Ω M (4 , − X , ω ), which indeed can have twodistinct parities according to [Boi15].Using the minimal holomorphic strata, we now construct both parities in the generalstrata of k -differentials of parity type for odd k . Let Ω k M g ( m , . . . , m n ) be a stratumof genus g ≥
2. Take a k -differential ( X , ξ ) in the stratum of genus zero differen-tials Ω k M ( m , . . . , m n , − kg ). We can construct a multi-scale k -differential ( X, ξ ) bygluing ξ to a primitive k -differential ( X , ξ ) in Ω k M g ( k (2 g − prim . According toLemma 5.7 and the case of the minimal strata above, the smoothing of such multi-scale k -differentials can give primitive k -differentials with both parities in the stratumΩ k M g ( m , . . . , m n ) prim if ( g, k ) = (2 , ξ with distinct parities. It remains to treat the strata in genus two for k = 3. Note that a stratum of cubicdifferentials is of parity type if and only if every singularity has even order. In whatfollows we deal with the strata of meromorphic cubic differentials of parity type ingenus two, and postpone the discussion of the remaining three holomorphic strata toSection 5.4.We start with the strata Ω M (2 n + 6 , − n ) prim . Take a multi-scale 3-differentialby gluing the zero of a 3-differential ( X , ξ ) in Ω M (2 n, − n ) prim with the pole ofa 3-differential ( X , ξ ) in Ω M (2 n + 6 , − n − prim . Such multi-scale 3-differentialsare always smoothable by the primitivity assumption on ξ and ξ . By Corollary A.18we can choose two ( X , ξ ) with distinct parities. Then after smoothing the multi-scale3-differentials we can obtain both parities in the stratum Ω M (2 n + 6 , − n ) prim byLemma 5.7.Next we consider the meromorphic strata Ω M (2 n + 6 , − ℓ , . . . , − ℓ s ) prim witha unique (analytic) zero. Take a multi-scale 3-differential by gluing the pole of a 3-differential ( X , ξ ) in Ω M (2 n + 6 , − n ) prim with the zero of a 3-differential ( X , ξ )in the genus zero stratum Ω M (2 n − , − ℓ , . . . , − ℓ s ). Since the top level differen-tial ξ is either primitive or contains a metric pole, such a multi-scale 3-differential issmoothable. By the preceding paragraph we can choose two ( X , ξ ) with distinct par-ities. Then after smoothing the multi-scale 3-differentials we thus obtain both paritiesin Ω M (2 n + 6 , − ℓ , . . . , − ℓ s ) prim by Lemma 5.7.Finally for the general meromorphic strata Ω M (2 n , . . . , n r , − ℓ , . . . , − ℓ s ) prim ,let n = P ri =1 n i . Take a multi-scale 3-differential by gluing the zero of a 3-differential( X , ξ ) in Ω M (2 n, − ℓ , . . . , − ℓ s ) prim with the pole of a 3-differential ( X , ξ ) inthe genus zero stratum Ω M (2 n , . . . , n r , − n − X , ξ ) with distinct parities. Then after smoothing the multi-scale3-differentials we thus obtain both parities in Ω M (2 n , . . . , n r , − ℓ , . . . , − ℓ s ) prim by Lemma 5.7.5.4. The strata Ω M (6) , Ω M (4 , and Ω M (2 , , . In this section we studythe remaining holomorphic strata of parity type in g = 2 and k = 3, which areΩ M (6), Ω M (4 ,
2) and Ω M (2 , , M (6) contains a hyperel-liptic component arising from the third power of abelian differentials in Ω M (2), andits complement Ω M (6) prim parameterizes primitive cubic differentials. On the otherhand, Ω M (4 ,
2) and Ω M (2 , ,
2) parameterize primitive cubic differentials only,as their zero orders are not divisible by three. Moreover, each of them contains ahyperelliptic component by Theorem 4.2.We can describe connected components of these strata as well as their parities ex-plicitly as follows.
Theorem 5.10.
The primitive stratum Ω M (6) prim is connected and has even parity.The strata Ω M (4 , and Ω M (2 , , both have two connected components, onebeing hyperelliptic and the other non-hyperelliptic. Moreover, Ω M (4 , hyp has oddparity and Ω M (4 , nonhyp has even parity, while Ω M (2 , , hyp has even parityand Ω M (2 , , nonhyp has odd parity. OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 29 Proof.
We first show that the stratum Ω M (6) prim is irreducible, and hence has aunique parity. Suppose ( X, ξ ) is in this stratum with divisor div ( ξ ) = 6 z ∼ K where K is the canonical class of X . Since X is hyperelliptic, the point z has ahyperelliptic conjugate which we denote by z ′ . Note that z ′ = z , for otherwise 2 z ∼ K would contradict the primitivity assumption. Since z + z ′ ∼ K , the previous conditionon z is equivalent to 3 z ∼ z ′ , i.e., K + 2 z ∼ z + z ′ ∼ z ′ . Consider the linear system | K + 2 z | which maps X to a plane quartic curve C . Since h ( X, K + z ) = h ( X, K ), theimage of z (still denoted by z for simplicity) is a cusp of C , and the cuspidal tangentline L at z cuts out 3 z + z ′ in C due to K + 2 z ∼ z + z ′ . Moreover, since 4 z ′ ∼ K + 2 z ,the tangent line to C at z ′ cuts out 4 z ′ (i.e. z ′ is a hyperflex). An example of suchcurves is illustrated in Figure 10 (where the coefficients a ij will be introduced later inthe proof). − − xy Figure 10.
A plane cuspidal quartic given by the choice of coefficients( a , a , a , a , a ) = (1 , , , , − x -axis) intersects the rest of the curve at a hyperflex.Conversely, suppose C is a plane cuspidal quartic satisfying that the cuspidal tangentline L at the cusp z cuts out 3 z + z ′ with z ′ = z and that the tangent line L ′ to C at z ′ cuts out the hyperflex 4 z . As long as C has no other singularities besides the cusp z ,we can recover ( X, ξ ) ∈ Ω M (6) prim (up to scale) by taking X to be the normalizationof C and z to be the unique zero of ξ . It is thus sufficient to show that the locus ofsuch special quartics C in the total space of plane quartics is irreducible.Let x and y be the affine coordinates of P . Without loss of generality, we can choose z = (0 , z ′ = (1 , L : y = 0 and L ′ : x − P . Let f ( x, y ) = X i + j ≤ a ij x i y j be the defining equation for a plane quartic curve C . In other words, the coefficients a ij give (homogeneous) coordinates for the parameter space P of plane quartics. Thecondition that C has a cusp (or a further degeneration) at z with L as the cuspidaltangent line is equivalent to that f belongs to the ideal generated by y and ( x, y ) ,i.e., a = a = a = a = a = 0 . The condition that L ′ ∩ C = 4 z ′ is equivalent to that f (1 , y ) is divisible by y , i.e., a + a = a + a = a + a + a = a + a = 0 . Note that if a = 0, then f ∈ ( x, y ) and consequently C would have a triple pointat z (i.e., a singularity worse than an ordinary cusp). Hence we can assume that up toscale a = 1. In this case f ( x, y ) reduces to f = y + a ( x − x ) + a ( x y − x y ) + a xy − ( a + 1) x y + a ( y − xy ) + a y which is parameterized by the five independent coefficients a , a , a , a , a . Onechecks that a generic choice of these parameters gives rise to a desired cuspidal curve(with no other singularities), whose normalization together with the cusp determines( X, z ) ∈ P Ω M (6) prim . We thus conclude that P Ω M (6) prim is the image of a denseopen subset of C = { ( a , a , a , a , a ) } , hence is irreducible. Alternatively, onechecks that the subgroup of the automorphism group of P that fixes z, z ′ , L, L ′ is3-dimensional, and since dim P Ω M (6) prim = 2, it implies that the locus of the cor-responding plane quartics must be 5-dimensional, thus filling a dense subset of theparameter space C .Similarly we can show that the non-hyperelliptic locus Ω M (4 , nonhyp is irre-ducible, thus giving rise to a connected component of the stratum. Suppose ( X, z , z )is contained in this locus with 4 z + 2 z ∼ K such that z and z are not Weier-strass points by the non-hyperelliptic assumption. The condition is equivalent to that K +2 z ′ ∼ z +2 z . In this case the linear system | K +2 z ′ | maps X to a plane quartic C with a cusp at z ′ such that the cuspidal tangent line L cuts out 3 z ′ + z with C andthe tangent line L to C at z cuts out 2 z + 2 z (i.e. L is a bitangent line). Using theabove affine coordinates, we can choose z ′ = (0 , z = (1 , z = (1 , L : y = 0and L : x − P . Then the coefficients ofthe defining equation f ( x, y ) = P i + j ≤ a ij x i y j of C satisfy that a = a = a = a = a = 0 ,a + a = a + a = 0 ,a + a + a − a = a + a + 2 a = 0 . As before we can assume that up to scale a = 1. In this case f ( x, y ) can be pa-rameterized by the five independent coefficients a , a , a , a , a . We thus concludethat the non-hyperelliptic locus in P Ω M (4 ,
2) is the image of a dense open subset of C = { ( a , a , a , a , a ) } , hence is irreducible. Alternatively, one checks that thesubgroup of the automorphism group of P that fixes z , z ′ , z , L , L is 2-dimensional,and since dim P Ω M (4 ,
2) = 3, it implies that the locus of the corresponding planequartics must be 5-dimensional, thus filling a dense subset of the parameter space C . OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 31 Next we show that the non-hyperelliptic locus Ω M (2 , , nonhyp is irreducible.Suppose ( X, z , z , z ) is contained in this locus with 2 z + 2 z + 2 z ∼ K such thatnone of the z i is a Weierstrass point by the non-hyperelliptic assumption. The conditionis equivalent to that K + z + z ∼ z ′ + z ′ +2 z ′ . In this case the linear system | K + z + z | maps X to a plane nodal quartic C where z and z coincide at the node. Moreover,the two tangent lines L and L to the two branches of the node respectively cut out2 z + z + z ′ and 2 z + z + z ′ in X , and the line L spanned by z ′ and z ′ is tangentto C at z ′ . We can choose z = z = (0 , z ′ = (1 , z ′ = (0 , z ′ = ( , ), L : y = 0, L : x = 0 and L : x + y − P .The coefficients of the defining equation f ( x, y ) = P i + j ≤ a ij x i y j of C satisfy that a = a = a = a = a = a + a = a + a = 0 ,a + a + a − ( a + a + a ) = 0 , a + a + a + a + 2( a + a + a ) = 0 . Up to scale there are five independent parameters. Moreover, the subgroup of the auto-morphism group of P that fixes the above choice of points and lines is 1-dimensional.Therefore, we conclude that P Ω M (2 , , nonhyp gives rise to a 4-dimensional con-nected component of the (projectivized) stratum.Now we show that the irreducible stratum Ω M (6) prim has even parity. The ideais to construct cubic differentials parameterized in this stratum by smoothing a multi-scale cubic differential formed by gluing the third power of an abelian differential in thestratum Ω M (0) with a primitive cubic differential in Ω M (6 , − prim which has azero 3-residue at the pole. Note that Ω M (6 , − prim has two connected componentsΩ M (6 , − and Ω M (6 , − , which parameterize respectively the difference z − p of the zero and the pole of the differentials ( E, ξ ) being respectively a primitive 6-torsionand a 3-torsion in the underlying elliptic curve E . By Lemma 5.8 and Remark 5.9 (for k = 3, n = 1 and a i = 1), the component Ω M (6 , − of rotation number two con-tains cubic differentials with zero 3-residue at the pole and with odd parity. We thusconclude that the irreducible stratum Ω M (6) prim arises from smoothing the afore-mentioned multi-scale cubic differential formed by using the component Ω M (6 , − ,and after smoothing the parity being even follows from Lemma 5.7 and the fact thatΩ M (0) has odd parity.Next we show that the stratum Ω M (4 ,
2) has both parities. A cubic differential inthe stratum Ω M (4 ,
2) can be obtained by smoothing the multi-scale cubic differentialobtained by gluing a cubic differential ( X , ξ ) in the stratum Ω M (2 , −
2) with a cubicdifferential ( X , ξ ) in the stratum Ω M (4 , −
4) at their poles. Since ξ and ξ areprimitive, the global 3-residue condition holds automatically. Moreover, Ω M (2 , − M (4 , − admits a double cover of P ramified at the zero and the pole, and hence the smoothing of the corresponding multi-scale cubic differential leads to the hyperelliptic component. Moreover by Theorem A.17the parities of Ω M (4 , − and Ω M (4 , − are odd and even respectively. Wethus conclude that Ω M (4 , hyp and Ω M (4 , nonhyp have odd and even parityrespectively by using Lemma 5.7. To obtain both parities in the stratum Ω M (2 , , M (4 ,
2) with distinct parities into twodouble zeros. According to Proposition 3.1 this operation is realizable and preserves(non-)hyperellipticity in this case.Finally we show that the (non-)hyperelliptic component of Ω M (2 , ,
2) has paritydifferent from the corresponding (non-)hyperelliptic component of Ω M (4 , P , ξ ) in the connect stratumΩ M (2 , , −
10) used in the preceding operation of breaking up the zero have oddparity, which is verified in the Appendix (see Proposition A.7 and Remark A.11). (cid:3)
Remark 5.11.
Although in the above proof we do not use the component Ω M (6 , − in which z − p is a primitive 6-torsion, to supplement our understanding, one can showthat every cubic differential in Ω M (6 , − has a nonzero 3-residue at the pole. Ifthis is not the case, then the irreducible stratum Ω M (6) prim would also arise fromsmoothing a multi-scale cubic differential formed by gluing the third power of an abeliandifferential in Ω M (0) with a cubic differential in Ω M (6 , − . Since the parity ofΩ M (6 , − is even by Theorem A.17, it would imply that the parity of Ω M (6) prim is odd, which contradicts Theorem 5.10.6. Adjacency of the strata of quadratic differentials
The notion of adjacency was used by [Boi15] for abelian differentials and by [Lan08]for quadratic differentials of finite area. In this section we extend their results to thecase of quadratic differentials of infinite area (i.e., allowing metric poles).First we extend the notion of adjacency to k -differentials for all k . Definition 6.1.
Let
C ⊂ Ω k M g ( µ ) and C ⊂ Ω k M g ( ν ) be two connected componentsof strata of k -differentials. We say that C is adjacent to C and denote it by C ⊃ C , ifthere is a k -differential in C which can be obtained by breaking up a metric zero of a k -differential in C .Meromorphic abelian differentials have flat geometric representations in terms of(broken) half-planes and half-cylinders as basic domains, as shown in Section 3.3of [Boi15], which provides a powerful tool in the study of meromorphic differentials.Below we show that the same type of basic domain decomposition holds for quadraticdifferentials with metric poles. Lemma 6.2.
Let q be a quadratic differential with at least one pole of order ≥ .Then q can be obtained by gluing (broken) half-planes and half-cylinders with polygonalboundaries.Proof. Up to multiplying q by a complex number of norm one (i.e., up to rotation),we can assume that q has no vertical saddle connections. Then by the description ofSection 11.4 of [Str84] every vertical trajectory is either an infinite line or a half rayemanating from a conical singularity. The vertical flow decomposes the surface intohalf-planes or infinite strips, each of which has a single conical point on every boundaryvertical line. Take a horizontal ray emanating from the boundary conical point of eachhalf-plane to cut these half-planes into -planes. Now the desired (broken) half-planes OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 33 are obtained by gluing the vertical rays in the boundary of the -planes. Finally, cuteach vertical infinite strips along the saddle connection joining the two conical pointson the boundary of the strip, and glue the resulting half-infinite strips to form thedesired half-cylinders according to adjacency of the vertical boundary rays. (cid:3) According to [Boi15, Proposition 6.1] each connected component of a stratum ofmeromorphic abelian differentials contains differentials that are obtained by the bub-bling operation. We generalize the result to the case of quadratic differentials.
Proposition 6.3.
Let µ = ( n, − l , . . . , − l s ) be a partition of g − with g ≥ , n ≥ , l i ≥ and at least one l j ≥ (i.e., with a unique analytic zero and at least one metricpole). Then every connected component C of the stratum Ω M g ( µ ) contains quadraticdifferentials obtained by bubbling a handle from a quadratic differential of genus g − .Proof. Let (
X, q ) be a quadratic differential in C . Up to rotation, we can assume that q does not admit vertical saddle connections, hence it has a basic domain decompositionby (broken) half-planes and half-cylinders as in Lemma 6.2. Local coordinates of C at q are given by 2 g + s − γ i at the boundary of the basic domains.We first show that the endpoints of each γ i cannot be two simple poles. Otherwise,since there is no vertical saddle connection, analyzing vertical trajectories in the basicdomain decomposition implies that this case can occur if and only if the stratum isΩ M ( − , − , − g ≥
1. Therefore, each γ i either joins the unique (analytic) zero z to itself or joins z to a simple pole. Sincethe span of all γ i contains the absolute homology H ( X, Z ), let γ , . . . , γ m be thoseboundary saddle connections joining z to itself which generate H ( X, Z ). The inter-section number between any two such closed paths is either 0 or ±
1. Since γ , . . . , γ m generate H ( X, Z ), there exist γ i and γ j such that their intersection number is ± γ i and γ j until they are very short compared to the other γ k which stay un-changed. Then a small neighborhood enclosing the saddle connections γ i and γ j isisometric to the complement of a neighborhood of the pole for a flat surface of genusone in Ω M ( n, − n ). To see it, note that in the shrinking process the zero remains tobe of order n . Since the intersection number of γ i and γ j is ±
1, the boundary of thisneighborhood is connected. It implies that the resulting differential has a unique pole.Since it has two saddle connections, the genus must be one.Summarizing the above discussion, we conclude that the limit of shrinking γ i and γ j to zero gives rise to a multi-scale 2-differential consisting of a quadratic differential ofgenus g − − n .We can further replace the (smooth) differential of genus one by a rational nodal differ-ential ( X , η ) as in Figure 2. Note that the resulting multi-scale 2-differential remainssmoothable even if the 2-residue at the separating node is nonzero, since the top levelcomponent of genus g − (cid:3) In [Boi15, Proposition 7.1] an adjacency property for the strata of meromorphicabelian differentials is described by merging zeros. We generalize the result to the caseof quadratic differentials.
Proposition 6.4.
Let C be a connected component of the stratum of quadratic differ-entials Ω M g ( n , . . . , n r , − a , − l , . . . , − l s ) with n i ≥ , l j ≥ and s > (i.e., with atleast one metric pole). Then for any ≤ b ≤ a , there exists a connected component C of Ω M g ( P ri =1 n i − b, − a − b , − l , . . . , − l s ) such that C ⊂ C . The notation − a stands for the signature of a simple poles. The above result saysthat we can merge all analytic zeros together with any specified number of simplepoles. We remark that it is possible that a primitive component C is adjacent to anon-primitive component C . Proof.
The claim obviously holds for g = 0, since any stratum of differentials of genuszero is connected and hence one can merge any two singularities. From now on weassume that g ≥
1. Let (
X, q ) be a quadratic differential in the component C . Upto rotation, we can assume that q does not have vertical saddle connections, hence q admits a basic domain decomposition given by Lemma 6.2. Since X is connected,there exist two (broken) half-plane or half-cylinder basic domains D and D such thatthey contain the same saddle connection γ joining two distinct metric zeros on theirboundary. Note that D and D are not necessarily distinct, and if they are identical,then γ appears twice on the boundary of the same basic domain.As we have seen in the proof of Propoisition 6.3, the endpoints of γ cannot be twosimple poles for g ≥
1. Next suppose the endpoints of γ consist of a simple pole p and an analytic zero z . Since the total angle at p is π , then γ appears twice on theboundary of the same basic domain D . If there is no other analytic zero besides z , wecan choose to shrink γ (or choose not to), and continue this process for the other simplepoles. If there are some other analytic zeros, then besides γ there must exist anothersaddle connection on the boundary of two basic domains. Iterate this procedure untilwe find a saddle connection joining two analytic zeros. Then we can shrink it to bearbitrarily short and consequently merge the two endpoint zeros. During the shrinkingprocess all the other boundary segments are fixed, hence the resulting surface remainsto be smooth and connected. By induction we can thus merge together all analyticzeros. Finally by the same argument as before, each of the remaining simple poles p i joins the totally merged zero by a boundary saddle connection γ i for i = 1 , . . . , a . Wecan choose to shrink b of the γ i for any b ≤ a , thus proving the result. (cid:3) Besides merging (metric) zeros, one can also merge poles together as given in thefollowing companion result.
Proposition 6.5.
Let C be a connected component of the stratum of quadratic differ-entials Ω M g ( n , . . . , n r , − a , − l , . . . , − l s ) with n i ≥ , l j ≥ and s > (i.e., with atleast one metric pole). Then for any ≤ b ≤ a , there exists a connected component C of the stratum Ω M g ( n , . . . , n r , − a − b , − P si =1 l i − b ) such that C ⊂ C . In other words, the above result says that we can merge all metric poles togetherwith any specified number of simple poles.
Proof.
Let (
X, q ) be a meromorphic quadratic differential in C . If q has at least twometric poles, then there exist two metric poles p and p such that they share a saddle OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 35 connection γ on the boundary of their basic domains D and D . Take a ray ℓ i ema-nating from the middle point of γ into D i such that ℓ i does not meet other singularitiesof q for i = 1 ,
2. Then we can stretch each side of γ to be arbitrarily long (i.e., pusheach side of D i far away from ℓ i ), which merges the poles p and p in the limit.Next suppose we have reduced q to have a unique metric pole p with some simplepoles z i for i = 1 , . . . , a . A metric neighborhood of z i is a half-disk with either aboundary ray or a saddle connection γ i . If γ i extends as a boundary ray of a half-plane, then the stratum is Ω M ( − , −
3) and there is nothing to prove. Suppose γ i isa saddle connection. Take a general ray ℓ i emanating from z i to p such that it does notmeet the other singularities of q . Cut the basic domain along ℓ i and push z i to infinityfrom both sides of ℓ i . Then z i and p are merged in the limit. We can choose b of the a simple poles and merge them one by one with p , thus proving the claim. (cid:3) Using the above results, we can show that the number of connected components ofa stratum of quadratic differentials does not increase when metric zeros are mergedtogether. This generalizes [Lan08, Corollary 2.7] to the case of quadratic differentialswith metric poles. Below we adapt the same notation from Proposition 6.4.
Proposition 6.6.
Let C and C be two connected components of the stratum of qua-dratic differentials Ω M g ( n , . . . , n r , − a , − l , . . . , − l s ) , and let C be a connected com-ponent of the stratum Ω M g ( n, − a − b , − l , . . . , − l s ) with n = n + · · · + n r − b . If C is contained in both C and C , then C = C . In particular, the number of connectedcomponents of Ω M g ( n , . . . , n r , − a , − l , . . . , − l s ) is bounded above by the number ofconnected components of Ω M g ( n, − a − b , − l , . . . , − l s ) .Proof. By assumption, there exist quadratic differentials in C i for each i = 1 , n of quadratic differentials in C into metric zerosof order n , . . . , n r , − b . Since C is connected, it suffices to check that the parameterspace of quadratic differentials in Ω M ( − n − k, n , . . . , n r , − b ) with a prong markingat the pole of order − n − k is connected, and this holds by Lemma 2.4. Moreover, byProposition 6.4 each connected component of Ω M g ( n , . . . , n r , − a , − l , . . . , − l s ) isadjacent to at least one connected component of Ω M g ( n, − a − b , − l , . . . , − l s ). Hencethe number of connected components of the former stratum is bounded by the latter. (cid:3) By a completely analogous argument we can obtain the following result for mergingpoles. We adapt the same notation from Proposition 6.5.
Proposition 6.7.
Let C and C be two connected components of the stratum of qua-dratic differentials Ω M g ( n , . . . , n r , − a , − l , . . . , − l s ) , and let C be a connected com-ponent of the stratum Ω M g ( n , . . . , n r , − a − b , − l ) with l = P sj =1 l j + b . If C iscontained in both C and C , then C = C . In particular, the number of connectedcomponents of Ω M g ( n , . . . , n r , − a , − l , . . . , − l s ) is bounded above by the number ofconnected components of Ω M g ( n , . . . , n r , − a − b , − l ) . Remark 6.8.
We expect that similar adjacency results by merging zeros or poleshold for the strata of k -differentials for general k . However, an analogue of verticaltrajectories and the resulting basic domain decomposition for (meromorphic) quadratic differentials seems not available for k -differentials in general, although some weak formof decomposition by using (broken) k -planes, half-infinite cylinders and some finite coreparts of the surface might exist. We leave it as an interesting question to investigatein future work. 7. Quadratic differentials with metric poles
In this section we focus on quadratic differentials with at least one metric pole, i.e.a pole of order at least 2. They correspond to half-translation surfaces of infinite area.Let µ = ( n , . . . , n r , − l , . . . , − l s ) be a partition of 4 g − − l i ,where at least one l i ≥
2. For notation simplicity, we denote by Q ( µ ) the correspondingstratum of quadratic differentials. Note that our definition of Q ( µ ) is slightly differentfrom the setting of [Lan08], as we also include quadratic differentials arising fromsquares of abelian differentials.We begin by introducing some notions to distinguish components of Q ( µ ). Let C be a connected component of Q ( µ ). In Section 4 we have introduced and classifiedthe hyperelliptic components. If C parameterizes squares of abelian differentials, wesay that C is of abelian type. If furthermore C parameterizes fourth powers of (non-hyperelliptic) theta characteristics (i.e. half-canonical divisors), then according to theirparity we say that C is of abelian-even or abelian-odd type. We will write hyp, ab, ab-even, and ab-odd for brevity. A (possibly disconnected) component which is not ofhyp (resp. ab) type is denoted by nonhyp (resp. nonab). These types characterize allpossible connected components of the strata Q ( µ ) for g ≥ Theorem 7.1.
Suppose Q ( µ ) is a stratum of quadratic differentials with genus ≥ and at least one metric pole. Then the following statements hold: (1a) If µ is (4 n, − l ) , (4 n, n, − l ) , (4 n, − l, − l ) , (4 n, n, − l, − l ) , (4 n, − , − ,or (4 n, n, − , − (except (8 , − , (4 , , − , (8 , − , − and (4 , , − , − ),then Q ( µ ) has four connected components, which are of hyp, ab-even, ab-odd,and nonab-nonhyp type. (1b) If µ is (8 , − , (4 , , − , (8 , − , − , or (4 , , − , − , then Q ( µ ) has threeconnected components, where the hyperelliptic component coincides with the ab-odd component in the first two cases and with the ab-even component in the lasttwo cases. (2) If µ is (2 n, − l ) , (2 n, n, − l ) , (2 n, − l, − l ) or (2 n, n, − l, − l ) , in all ofwhich l > and n, l are not both even, or if µ is (2 n, n, − , − with n odd,then Q ( µ ) has three connected components, which are of hyp, ab, and nonab-nonhyp type. (3) If µ is (4 n , . . . , n r , − l , . . . , − l s ) with r ≥ or s ≥ , or (4 n , n , − l , − l ) with n = n or l = ℓ , or (4 n, − l , − l ) with l = l , or (4 n , n , − l ) with n = n , or (4 n , . . . , n r , − , − with r ≥ , or (4 n , n , − , − with n = n , then Q ( µ ) has three connected components, which are of ab-even,ab-odd, and nonab-nonhyp type. OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 37 (4) If µ is (2 n, − l, − l ) , ( n, n, − l ) or ( n, n, − l, − l ) , in all of which n and l areboth odd, or if µ is (2 n, − or (2 n, n, − , then Q ( µ ) has two connectedcomponents, which are of hyp and nonhyp type. (5) If µ is either (2 n , . . . , n r , − l , . . . , − l s ) with s ≥ or r ≥ (except thepartitions (2 n , . . . , n r , − ), or (2 n , n , − l , − l ) with n = n or l = l ,or (2 n, − l , − l ) with l = l , or (2 n , n , − l ) with n = n and l > , inall of which n i and l j are not all even, or if µ is (2 n , . . . , n r , − , − with n i not all even, then Q ( µ ) has two connected components, which are of ab andnonab-nonhyp type. (6) If µ is non of the above, i.e., µ contains at least one odd entry and is not oftype (4) , or µ is of the form (2 n , . . . , n r , − and not of type (4) , then Q ( µ ) is connected (and is of nonab-nonhyp type). To prove the theorem we will follow closely the strategy in [Boi15] and remark oncomparable results. Let us first review some notations and preparations.Recall that a quadratic differential in Q ( µ ) corresponds to a half-translation surfacethat has a basic domain decomposition as in [Boi15], where parallel edges can beidentified by reflection besides translation, including the half-infinite boundary rays ofeach basic domain. Note that simple poles of a quadratic differential correspond toconical singularities of angle π under the induced flat metric, and in our notation theyare both analytic poles and metric zeros. In this section we will view these singularitiesas analytic poles with one exception, which is in genus zero when the singularity ofhighest order is a simple pole but viewed as a metric zero. In this sense we alwayshave at least one metric zero for every stratum Q ( µ ). If r = 1, i.e., when there is aunique analytic zero, we say that Q ( µ ) is a minimal stratum . Note that Proposition 6.3implies that any minimal stratum contains a half-translation surface obtained from theoperation of bubbling a handle.We begin the proof of Theorem 7.1 by bounding the number of connected componentsof the minimal strata. Proposition 7.2 ([Boi15, Proposition 6.2]) . Let Q ( n, − l , . . . , − l s ) be a minimal stra-tum of quadratic differentials in genus g ≥ with at least one l i ≥ . Then the followingstatements hold: (i) If n is odd, then Q ( n, − l , . . . , − l s ) is connected. (ii) If n is even, then Q ( n, − l , . . . , − l s ) has at most four connected components. (iii) If n is even and at least one l i is odd, then Q ( n, − l , . . . , − l s ) has at most twoconnected components.Proof. Let C be a connected component of Q ( n, − l , . . . , − l s ). By Proposition 6.3, if Q ( n, − l , . . . , − l s ) = Q (4 g − , − s , . . . , s g such that C = C ⊕ s ⊕ · · · ⊕ s g , where C is the connected stratum Q ( n − g, − l , . . . , − l s ) of quadratic differentials ofgenus zero, and 1 ≤ s i ≤ n − g + 4 i −
1. In the case of Q (4 g − , − s , . . . , s g − such that C = C ⊕ s ⊕ · · · ⊕ s g − , Label µ n, − l ), (4 n, n, − l, − l ), hyp(1a) (4 n, − l, − l ), (4 n, n, − l ), 4 ab-even(4 n, − , − n, n, − , − , − , , − , − , − , , − , −
2) 3 hyp = ab-even, ab-oddnonab-nonhyp(2 n, − l ), (2 n, − l, − l ), (2 n, n, − l, − l ), hyp(2) in all of which l > n, l are not both even, 3 ab(2 n, n, − , −
2) with n odd nonab-nonhyp(4 n , . . . , n r , − l , . . . , − l s ) with r ≥ s ≥ n , n , − l , − l ) with n = n or l = ℓ , ab-even(3) (4 n, − l , − l ) with l = l , 3 ab-odd(4 n , n , − l ) with n = n , nonab-nonhyp(4 n , . . . , n r , − , −
2) with r ≥ n , n , − , −
2) with n = n (2 n, − l, − l ) with l odd,(4) ( n, n, − l ) with n odd, 2 hyp( n, n, − l, − l ) with n and l not both even, nonhyp(2 n, − n, n, − n , . . . , n r , − l , . . . , − l s ) with s ≥ r ≥ n , . . . , n r , − n , n , − l , − l ) with n = n or l = l , ab(5) (2 n, − l , − l ) with l = l , 2 nonab-nonhyp(2 n , n , − l ) with n = n and l > n i and l j are not all even,(2 n , . . . , n r , − , −
2) with n i not all even µ has at least one odd entry,(6) (2 n , . . . , n r , − Table 1.
Connected components of the strata of quadratic differentialswith at least one metric pole.where C is the connected stratum Q (2 , −
2) of quadratic differentials in genus one, and1 ≤ s i ≤ i + 1.By Corollary 3.10 we can assume that s i ≤ s i +1 for all i . By Proposition 3.8 (iv) wecan further assume that s i ≤ s i +1 ≤ s i + 3. Next suppose two adjacent s i and s i +1 arenot of balanced type (see Definition 3.9). If s i ≥
3, then apply Proposition 3.8 (ii), (iii),(iv) and (ii) (in this order) so that we can reduce s i ⊕ s i +1 to ( s i − ⊕ ( s i +1 − s i ≤ i < g and s g ≤ s g − + 3. In addition, Proposition 3.8(iii) implies that 1 ⊕ ⊕ ⊕ ⊕
4, where the latter is further equal to
OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 39 ⊕ Q (2 , − ⊕ Q (2 , − ⊕ C is given by one of the following cases:(1) C ⊕ ⊕ · · · ⊕ ⊕ ⊕ · · · ⊕ C ⊕ ⊕ · · · ⊕ ⊕ ⊕ · · · ⊕ ⊕ C ⊕ ⊕ · · · ⊕ ⊕ ⊕ · · · ⊕ ⊕ C ⊕ ( n − g + 2) ⊕ ( n − g + 4) ⊕ · · · ⊕ n if g ≥ n is even,where the numbers of 1 and 2 in the sequences are allowed to be zero and the last casecorresponds to the balanced type in Definition 3.9. We remark that the zero order n in the above differs from the one used in Proposition 3.8, as herein the zero order webubble with parameter s i varies with the index i in each step.Consider first when n is odd. Note that in this case the only balanced type is( s , s ) = (1 ,
3) when we bubble a metric zero of order − k = 2 and n = − Q ( − , − , − Q ( − , − ⊕ ⊕ Q (7 , − , −
2) and Q (7 , −
3) in genus two that will be treated separately. Since n is odd,we can assume, using the gcd-trick of Lemma 3.14, that there is no ⊕ C ⊕ ⊕ · · · ⊕ . For case (2), Proposition 3.8 (i) allows us to change ⊕ ⊕ ( n −
3) as long as C is not one of the two connected strata Q ( − , − , −
2) and Q ( − , −
3) (as applying ⊕ s to a metric zero of order − s ≤ , l , . . . , l s ) =gcd(2 , l , . . . , l s ) = 1 (as n being odd implies that some l i must be odd), hence case (3)reduces to case (1). It remains to prove that the two special strata Q (7 , − , −
2) and Q (7 , −
3) are connected. By Corollary 2.6 the loci of multi-scale 2-differentials obtainedby identifying the marked zero of a quadratic differential respectively in Q (3 , − , − Q (3 , −
3) to the marked pole of a quadratic differential in Q (7 , −
7) are connected.Using Proposition 6.3, this implies that Q (7 , − , −
2) and Q (7 , −
3) are connected. Insummary, we conclude that for n odd and g ≥ Q ( n, − l , . . . , − l s ) can be obtained by the operation ⊕ ⊕ · · · ⊕
1, and hence Q ( n, − l , . . . , − l s ) is connected in this case, thus proving part (i) of Proposition 7.2.Next consider when n is even and g ≥
2. We first discuss the balanced types. Notethat the balanced types ( s , s ) = (1 ,
4) and (2 ,
4) (when we bubble an ordinary point,i.e., a zero of order zero) are special cases of (3). Moreover, the balanced type C ⊕ ( a + 2) ⊕ ( a + 4) ⊕ · · · ⊕ ( a + 2 g )with a = n − g cannot be reduced in general by using Proposition 3.8. The strata ofgenus zero that parameterize quadratic differentials with a zero of order zero and atleast one metric pole are the following(7.1) Q (0 , − , Q (0 , − , − , Q (0 , − , −
3) and Q (0 , − , − , − . We would like to break the balanced type C ⊕ ⊕ C being any one of the abovefour strata and break the balanced type C ⊕ ⊕ C being any one of the last two of the four strata. Since the last two strata listed in (7.1) both contain a singularityof odd order, we can break the balanced type ⊕ ⊕ ⊕ ⊕ C ⊕ ( a + 2) ⊕ ( a + 4) ⊕ · · · ⊕ ( a + 2 g )with a = n − g can be reduced to one of the cases (1) to (3).From now on we consider when n is even and the direct sum is not of balanced type,i.e., cases (1) to (3). First assume that n ≡ ⊕ ⊕ ⊕
3, hence it reduces to case (2). Moreover, changing ⊕ ⊕ ( n − C ⊕ ⊕ · · · ⊕ ⊕ ⊕ · · · ⊕ , C ⊕ ⊕ · · · ⊕ ⊕ . Next consider the case when n ≡ ⊕ ⊕ ( n − ⊕ ⊕ ⊕ C ⊕ ⊕ · · · ⊕ ⊕ ⊕ · · · ⊕ , C ⊕ ⊕ · · · ⊕ ⊕ . If n is even and at least one l i is odd, then gcd( s , l , . . . , l s ) = gcd(2 s , l , . . . , l s ).Hence we have C ⊕ C ⊕ C ⊕ , C ⊕ ⊕ C ⊕ ⊕ C ⊕ ⊕ , where we used the gcd-trick and the last equality follows from applying Proposition 3.8(ii) and (iv). It implies that in this case we only need to consider C ⊕ ⊕ · · · ⊕ n and all l i are even. We claim that in this case C ⊕ ⊕ C ⊕ ⊕ C ⊕ ⊕ . This claim can be checked by degenerating quadratic differentials in each connectedcomponent in the above to a multi-scale 2-differential on a 2-nodal union of a genusone curve X with a rational curve X (see Figure 11), in such a way that the locusof such multi-scale 2-differentials is irreducible. We will first prove the claim for thosestrata with a unique metric pole. Then for strata with arbitrary numbers of analyticpoles the claim follows from combining Propositions 6.5 and 6.7 that deal with merginganalytic poles (including at least one metric pole) to a single metric pole.More precisely, a quadratic differential in Q (2 m, − m − ⊕ ⊕ Q (2 m, − m − ⊕ ⊕ OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 41 zX p X q q Figure 11.
The pointed curve underlying the multi-scale 2-differentialused in the proof of C ⊕ ⊕ C ⊕ ⊕ C ⊕ ⊕ n andall l i are even.represented by 12123434 with the difference that the segments labeled by 3 and 4 areidentified by translation as shown on the right of Figure 12.1 2 1 2 3 4 3 45 65 6 1 2 1 23 44 35 65 6 Figure 12.
Quadratic differentials in Q (2 m, − m − ⊕ ⊕ Q (2 m, − m − ⊕ ⊕ m = 0.We first consider the special stratum Q (6 , − Q (2 , − ⊕ Q (2 , − ⊕ Q (2 , −
2) cannot be further realized as bubbling Q ( − , −
2) because − Q ( − , , −
1) and a differential in Q (6 , − , − , −
7) and ( − , −
3) are glued together respectively to form the two nodes. Accordingto Corollary 2.7 such multi-scale 2-differentials have a unique 2-prong-matching, hencethis locus is irreducible. The fact that the moduli space of multi-scale 2-differentials issmooth implies that Q (2 , − ⊕ Q (2 , − ⊕ Q (2 m, − m −
4) with m ≥
0. Inboth cases Q (2 m, − m − ⊕ ⊕ Q (2 m, − m − ⊕ ⊕ Q ( − m − , m + 5 , −
1) in genus oneand Q (2 m + 8 , − m − , −
3) in genus zero respectively. It follows from Corollary 2.7that the locus of such multi-scale 2-differentials is irreducible. Since the moduli spaceof multi-scale 2-differentials is smooth, we conclude that Q (2 m, − m − ⊕ ⊕ Q (2 m, − m − ⊕ ⊕ It remains to prove that Q (2 m, − m − ⊕ ⊕ Q (2 m, − m − ⊕ ⊕ m ≥ ⊕ ⊕ Q (2 m, − m − ⊕ ⊕ Q (2 m, − m − ⊕ ⊕ C ⊕ ⊕ C ⊕ ⊕ C = Q (0 , − ⊕ ⊕ Q (5 , − , −
4) and the otherin Q (8 , − , − ⊕ ⊕ ′ on the left) to obtainthe new representation of this quadratic differential on the right of Figure 13. Thenwe shrink the segment labeled by 3 ′ to zero. One checks that this degeneration leadsto the same locus of multi-scale 2-differentials as before. It follows from Corollary 2.7that this locus is irreducible, thus proving that Q (0 , − ⊕ ⊕ Q (0 , − ⊕ ⊕ ′ ○ ′ ′ ○ Figure 13.
The deformation in the case Q (0 , − ⊕ ⊕ ′ for quadratic differentials obtained from ⊕ ⊕
4. Then a sim-ilar construction leads to the degenerations to a banana curve with one irreduciblecomponent in Q (5 , − , − , − Q (5 , − , − , −
2) or Q (5 , − , − , − , − Q (8 , − , − C ⊕ ⊕ C ⊕ ⊕ C ⊕ ( n − g + 2) ⊕ ( n − g + 4) ⊕ · · · ⊕ n , C ⊕ ⊕ · · · ⊕ , C ⊕ ⊕ · · · ⊕ ⊕ , C ⊕ ⊕ · · · ⊕ , thus verifying part (ii) of Proposition 7.2. (cid:3) OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 43 Remark 7.3.
The above proof indeed gives more information compared to the state-ment of Proposition 7.2, as it describes explicitly the possible bubbling operations thatgive rise to various connected components of the strata.We next show that the existence of suitable pole orders can eliminate some of thepotential connected components listed in Proposition 7.2.
Proposition 7.4 ([Boi15, Proposition 6.3]) . Let Q ( n, − l , . . . , − l s ) be a minimal stra-tum of quadratic differentials of genus g ≥ with s ≥ , n and all l i even and at leastone l i ≥ . Let C denote the (connected) genus zero stratum Q ( n − g, − l , . . . , − l s ) .Then the following statements hold: (1) If some l i is not divisible by and P si =1 l i > , then C ⊕ ⊕ · · · ⊕ C ⊕ ⊕ · · · ⊕ ⊕ . (2) If s > or l = l , then C ⊕ ( n − g + 2) ⊕ ( n − g + 4) ⊕ · · · ⊕ n = C ⊕ ⊕ · · · ⊕ ⊕ t for some t ∈ { , } .Proof. The upshot behind this result is that under the assumption of (1) the ab-paritytype is not available and under the assumption of (2) the hyperelliptic type is not avail-able. The proof is the same as [Boi15, Proposition 6.3] by multiplying each summandin the direct sums therein by two. (cid:3)
Using Propositions 7.2 and 7.4 we can classify the connected components of theminimal strata of quadratic differentials, which generalizes [Boi15, Theorem 6.4] to thecase of quadratic differentials and proves Theorem 7.1 in the case of the minimal strata.
Theorem 7.5.
Let µ be a signature of quadratic differentials for g ≥ with a uniqueanalytic zero and at least one metric pole. Then the connected components of theminimal stratum Q ( µ ) can be described as follows: (1a) If µ is (4 n, − l ) , (4 n, − l, − l ) , or (4 n, − , − (except (8 , − and (8 , − , − ),then Q ( µ ) has four connected components, which are of hyp, ab-even, ab-odd,and nonab-nonhyp type. (1b) If µ is (8 , − or (8 , − , − , then Q ( µ ) has three connected components, wherethe hyperelliptic component coincides with the ab-odd component in the firstcase and with the ab-even component in the second case. (2) If µ is (2 n, − l ) with l > odd or (2 n, − l, − l ) with l > odd, then Q ( µ ) hasthree connected components, which are of hyp, ab, and nonab-nonhyp type. (3) If µ is (4 n, − l , . . . , − l s ) with s ≥ or (4 n, − l , − l ) with l = l , then Q ( µ ) has three connected components, which are of ab-even, ab-odd, and nonab-nonhyp type. (4) If µ is (2 n, − l, − l ) with l odd or (2 n, − , then Q ( µ ) has two connected compo-nents, which are of hyp and nonhyp type. (5) If µ is (2 n, − l , . . . , − l s ) with s ≥ or (2 n, − l , − l ) with l = l , in bothof which n, l , . . . , l s are not all even, then Q ( µ ) has two connected components,which are of ab and nonab-nonhyp type. (6) If µ is non of the above, i.e., µ contains at least one odd entry and is not oftype (4) , then Q ( µ ) is connected (and is of nonab-nonhyp type).Proof. If the unique analytic zero of µ has odd order, the claim follows from Propo-sition 7.2 (i). Hence we can assume that the unique zero has even order. Let C bea connected component of Q ( µ ) and C be the connected genus zero stratum of qua-dratic differentials Q (2 n − g, − l , . . . , − l s ). As pointed out in Remark 7.3, the proofof Proposition 7.2 shows that C is given by one of the following four cases:(a) C = C ⊕ ( n − g + 2) ⊕ ( n − g + 4) ⊕ · · · ⊕ n, (b) C = C ⊕ ⊕ · · · ⊕ , (c) C = C ⊕ ⊕ · · · ⊕ ⊕ , (d) C = C ⊕ ⊕ · · · ⊕ . When µ is (4 n, − l ), (4 n, − l, − l ) or (4 n, − , −
2) (except (8 , −
4) and (8 , − , − ⊕ s contributes s + 1 (mod 2) to the parity(see [KZ03, Lemma 11]). Case (d) corresponds to a connected component of primitivequadratic differentials, as the operation ⊕ π . Wehave thus verified (1a).For µ = (8 , −
4) in g = 2, cases (a) and (c) are both equal to C ⊕ ⊕
4. Let z be the zero and p be the pole. In the hyperelliptic component given by (a), both z and p are Weierstrass points in X . Since h ( X, z − p ) = h ( X, p ) = 1 is odd, in thiscase the hyperelliptic component coincides with the ab-odd component. Similarly for µ = (8 , − , −
2) in g = 2, cases (a) and (c) are both equal to C ⊕ ⊕
4. Let z be thezero and p , p be the two poles. Recall that the abelian parity in this case arises fromgluing p , p as a node to form an irreducible nodal curve X ′ of (arithmetic) genusthree. If z is a Weierstrass point in X and p , p are hyperelliptic conjugates (i.e.,2 z ∼ p + p in X ), we have h ( X ′ , z ) = 2 is even, hence in this case the hyperellipticcomponent coincides with the ab-even component. We have thus verified (1b).When µ is (2 n, − l ) or (2 n, − l, − l ), both with l > C ⊕ ⊕ · · · ⊕ C ⊕ ⊕ · · · ⊕ ⊕ , hence Q ( µ ) has at most three connected components. It is easy to see that this op-eration gives a connected component which is different from the components given bythe other two operations. More precisely, case (a) corresponds to hyp, case (b)=(c)corresponds to ab, and case (d) corresponds to nonab, thus verifying (2).When µ is (4 n, − l , . . . , − l s ) with s ≥ s = 2 and l = l , by Proposition 7.4(2) we have C ⊕ (2 n − g + 2) ⊕ (2 n − g + 4) ⊕ · · · ⊕ n = C ⊕ ⊕ · · · ⊕ ⊕ t for some t ∈ { , } . Hence Q ( µ ) has at most three connected components. It is easyto see that there are three connected components, as case (b) corresponds to ab-even,case (c) corresponds to ab-odd, and case (d) corresponds to nonab, thus verifying (3). OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 45 When µ = (2 n, − l, − l ) with l ≥ , l, l ) = gcd(2 , l, l ) = 1, hence we can further identifycase (b) with case (d) by Lemma 3.14. Therefore, if l ≥ Q (2 n, − l, − l )has at most two connected components. It is easy to see that both cases (a) and (d)occur, which correspond to hyp and nonhyp respectively.The case µ = (2 n, −
2) is special since we start the bubbling operation from the(connected) stratum Q (2 , −
2) of genus one (not zero) . We have checked in the proofof Theorem 7.2 that Q (2 , − ⊕ Q (2 , − ⊕
2, both giving the same non-hyperellipticconnected component of Q (6 , − Q (2 n, −
2) has at most two connected components. Indeed, case (a) is hyp and case (d)is nonhyp, hence Q (2 n, −
2) has exactly two connected components. Combining thiswith the preceding paragraph thus verifies (4).When µ = (2 n, − l , . . . , − l s ) with n, l , . . . , l s not all even and s ≥ s = 2 but l = l , it implies that some l i is odd, hence Proposition 7.4 (1) and (2) both apply. Itfollows that Q ( µ ) has at most two connected components. It is easy to see that in thiscase there are exactly two connected components corresponding to nonab in case (d)and ab in the other cases, thus verifying (5).Finally if the unique zero is of even order and we are not in one of the previous cases,then at least one analytic pole is of odd order, which implies thatgcd(1 , l , . . . , l s ) = gcd(2 , l , . . . , l s ) = gcd(4 , l , . . . , l s ) = 1 . Moreover, in this case we have s ≥ s = 2 but l = l . Hence combining Lemma 3.14with Proposition 7.4 implies that all the cases (a)–(d) give rise to the same connectedcomponent, thus completing the proof of (6). (cid:3) We now use Theorem 7.5 and the results in Section 6 to study non-minimal strata ofgenus g ≥
2. We first bound the number of connected components of a general stratumby using adjacency to the corresponding minimal stratum, which generalizes [Boi15,Proposition 7.2] to the case of quadratic differentials.Next we construct paths in the closure of certain strata that join a hyperellipticcomponent with a non-hyperelliptic component in the boundary strata, which general-izes [Boi15, Proposition 7.3 (2)].
Proposition 7.6.
Let Q = Q (2 n, − l ) (resp. Q (2 n, − l, − l ) ) be a genus g ≥ minimalstratum that contains a hyperelliptic component. For any n = n satisfying the equa-tion n + n = 2 n , there exists a path γ ( t ) ∈ Q ( n , n , − l ) (resp. Q ( n , n , − l, − l ) )such that γ (0) is in the hyperelliptic component of Q and γ (1) is in a non-hyperellipticcomponent of Q . We remark that the closure notation herein means taking the closure of a stratumin the corresponding moduli space of multi-scale 2-differentials.
Proof.
By Lemma 4.3 the hyperelliptic component of Q can be obtained as C ⊕ n where C is a hyperelliptic component in genus g −
1, and any component given by
C ⊕ s with s = n is non-hyperelliptic. The idea of the proof is to use this fact and reduce tothe case of genus one. The hyperelliptic component Q (2 n − , − n ) ⊕ n of the stratum Q (2 n, − n ) in genusone has rotation number gcd(2 n, n ) = n . Break up the metric zero of order 2 n to twometric zeros of order n and n . Since n + n = 2 n and n i = n for i = 1 ,
2, we havegcd( n , n , n ) = s < n . Therefore, as shown in the proof of Theorem 3.12 there existsa path in Q ( n , n , − n ) (in the closure of the component of rotation number s ) thatjoins the component Q (2 n − , − n ) ⊕ n to the component Q (2 n − , − n ) ⊕ s .Now fix a quadratic differential ( X g − , η g − ) in the hyperelliptic component of thestratum Q (2 n − , − l ) in genus g −
1. For a quadratic differential ( X , η ) ∈ Q (2 n, − n )in genus one, we construct a multi-scale 2-differential by gluing the pole of η to thezero of η g − and putting the unique equivalence class of 2-prong-matchings at thenode. After smoothing this multi-scale 2-differential we obtain a quadratic differen-tial ( X, η ) ∈ Q (2 n, − l ). Similarly we can carry out the same construction by using( X , η ) ∈ Q ( n , n , − n ) and obtain a quadratic differential ( X, η ) ∈ Q ( n , n , − l )after smoothing. Therefore, the desired path can be obtained by smoothing the cor-responding path in genus one, constructed in the preceding paragraph, that joins aquadratic differential ( X , η ) in the hyperelliptic component to a quadratic differen-tial ( X , η ) in a non-hyperelliptic component.The case Q (2 n, − l, − l ) is completely analogous. (cid:3) Next we construct paths in the closure of certain strata that join a component ofab-even type with a component of ab-odd type in the boundary strata.
Proposition 7.7 ([Boi15, Proposition 7.3 (1)]) . Let Q = Q (4 n, − l , . . . , − l s ) (resp. Q (4 n, − , − ) be a genus g ≥ minimal stratum. For any n , n satisfying the equality n + n = 4 n and not divisible by , there exists a path γ ( t ) ∈ Q ( n , n , − l , . . . , − l s ) (resp. Q ( n , n , − , − ) such that γ (0) is in the ab-even component of Q and γ (1) isin the ab-odd component of Q .Proof. The same proof as in [Boi15] works by multiplying each summand of the bub-bling operations therein by two. (cid:3)
Similarly we construct paths in the closure of certain strata that join a componentof ab-nonhyp type with a component of nonab type in the boundary strata.
Proposition 7.8.
Let Q = Q (2 n, − l , . . . , − l s ) be a genus g ≥ minimal stratumsuch that if s = 1 then l > . For any n , n odd with n + n = 2 n , there existsa path γ ( t ) ∈ Q ( n , n , − l , . . . , − l s ) such that γ (0) is in an ab-nonhyp componentof Q and γ (1) is in a nonab component of Q .Proof. Let C = Q (2 n − g, − l , . . . , − l s ) be the corresponding minimal stratum ofgenus zero. Then the connected components of Q given by C ⊕ ⊕ · · · ⊕ ⊕ C ⊕ ⊕ · · · ⊕ ⊕ C g − ⊕ C g − ⊕
1, where C g − = C ⊕ ⊕ · · · ⊕ Q (2 n − , − l , . . . , − l s ) in genus g − X g − , η g − ) in C g − . For any ( X , η ) ∈ Q (2 n, − n )in genus one, we can smooth the multi-scale 2-differential obtained by gluing the zeroof η g − to the pole of η with the unique 2-prong-matching at the node. This way weobtain a quadratic differential ( X, η ) in the stratum Q (2 n, − l , . . . , − l s ). Similarly we OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 47 can carry out the same procedure for ( X ′ , η ′ ) ∈ Q ( n , n , − n ) and obtain a quadraticdifferential ( X ′ , η ′ ) ∈ Q ( n , n , − l , . . . , − l s ), where ( X ′ , η ′ ) arises from breaking upthe zero of ( X , η ) into two zeros of order n and n respectively.Note that a quadratic differential in the connected component Q (2 n − , − n ) ⊕ Q (2 n, − n ) has rotation number gcd(2 n,
2) = 2. After breaking up thezero of order 2 n into two zeros of odd order n and n , the rotation number becomesgcd( n , n ,
2) = 1. Hence there is a path in Q ( n , n , − n ) that joins Q (2 n − , − n ) ⊕ Q (2 n − , − n ) ⊕
1. Using this path, the existence of the desired path in the genus g stratum Q ( n , n , − l , . . . , − l s ) thus follows from the construction described in thepreceding paragraph. (cid:3) After all these preparations, we can finally classify the connected components of thestrata of quadratic differentials in general.
Proof of Theorem 7.1.
Denote by Q = Q ( n , . . . , n r , − l , . . . , − l s ) a stratum of qua-dratic differentials of genus g ≥
2, with at least one l i ≥
2. Denote by Q min = Q ( n, − l , . . . , − l s )the corresponding minimal stratum, where n = n + · · · + n r . By Proposition 6.6 thenumber of connected components of Q is bounded above by the number of connectedcomponents of Q min .If n is odd, i.e., if l + · · · + l s is odd, then Q min is connected by Proposition 7.2,hence Q is connected.From now on we assume that n is even. Let us recall some related terminologyfirst. We say that the set of zero orders (resp. pole orders) is of hyperelliptic type ifit is { n, n } or { n } (resp. {− l, − l } or {− l } ), i.e., it is the set of zero orders (resp.pole orders) of a hyperelliptic component. We say that the partition is of abelian typeif every entry is even (and in the case where s = 1 then we require l > µ is ( − , −
2) (except for µ = (8 , −
4) or (8 , − , −
2) which we have treated already inTheorem 7.5 and for µ = (4 , , −
4) or (4 , , − , −
2) which will be treated separately atthe end of the proof), we say that it is of abelian-parity type, because the correspondingstratum of abelian differentials contains connected components of spin parity type,respectively. We now give a lower bound on the number of connected components.First consider the following cases: • If the stratum is Q ( n, n, − l ) (resp. Q ( n, n, − l, − l )), then it contains a hy-perelliptic component. The corresponding minimal stratum Q (2 n, − l ) (resp. Q (2 n, − l, − l )) contains one hyperelliptic component and at least one non-hyperelliptic component. Breaking up the zero of order 2 n in a non-hyperellipticquadratic differential gives a quadratic differential in a non-hyperelliptic com-ponent. Hence the stratum Q ( n, n, − l ) (resp. Q ( n, n, − l, − l )) contains onehyperelliptic component and at least one non-hyperelliptic component. • If the stratum is of abelian type, then the corresponding minimal stratum hasat least two non-hyperelliptic components, at least one of which arises from squares of abelian differentials (or more if it is also of abelian-parity type), andhas possibly a hyperelliptic component if it is in addition of hyperelliptic type.We can use the above discussion to give a lower bound on the number of connectedcomponents of Q . If the set of zero and pole orders of Q is of both hyperellipticand abelian-parity type (hence of abelian type), then Q has at least four connectedcomponents: hyp, ab-odd, ab-even, and nonab-nonhyp. This implies that such strata,described in Theorem 7.1 (1a), have exactly four connected components, as the lowerand upper bounds on the number of connected components coincide in this case (bothequal to four). Moreover, if the set of zero and pole orders of Q is of abelian-paritytype, then Q has at least three connected components: nonab, ab-odd, and ab-even.Similarly if it is of both abelian and hyperelliptic types, then Q has at least threeconnected components: hyp, ab, and nonab-nonhyp. Finally, if the set of zero and poleorders is of hyperelliptic or abelian type, then Q has at least two connected components.Next we give a refined upper bound for the number of connected components of Q as follows:(1) Suppose the set of poles of Q is of both hyperelliptic and abelian-parity types.In this case the minimal stratum has four connected components which we de-note by Q hypmin , Q nonabmin , Q ab-evenmin and Q ab-oddmin . Let C hyp , C nonab , C ab-even and C ab-odd be the connected components of Q that are adjacent to the respective connectedcomponents of the minimal stratum.(a) Suppose that the set of zeros of Q is not of hyperelliptic type, we can choosean entry n j such that n j = P i = j n i . Note that for any 1 ≤ j ≤ r , the stratum Q ( n j , P i = j n i , − l , . . . , − l s ) is adjacent to Q min . By Proposition 7.6 there is apath in Q ( n j , P i = j n i , − l , . . . , − l s ) joining Q hypmin to a non-hyperelliptic compo-nent of Q min . Breaking up the zero of order P i = j n i along this path into zerosof order ( n i ) i = j , we obtain a path in Q that joins a neighborhood of Q hypmin to aneighborhood of a non-hyperelliptic component of Q min . Hence C hyp coincideswith one of the non-hyperelliptic components C nonab , C ab-even or C ab-odd . Itfollows that in this case the number of connected components of Q is at mostthree.(b) If the set of zeros of Q is not of abelian-parity type, there exists an n j not divis-ible by four. By Proposition 7.7 there is a path in Q ( n j , P i = j n i , − l , . . . , − l s )joining Q ab-evenmin to Q ab-oddmin . Hence C ab-even = C ab-odd , and in this case thenumber of connected components of Q is at most three.(c) Suppose the set of zeros of Q is not of abelian type (hence not of abelian-paritytype). We have seen that C ab-even = C ab-odd . Moreover, we can choose an n j which is odd. By Proposition 7.8 there is a path in Q ( n j , P i = j n i , − l , . . . , − l s )joining Q abmin to Q nonabmin , which implies that C nonab coincides with one of C hyp and C ab-even = C ab-odd . Hence in this case the number of connected componentsof Q is at most two.(d) If the set of zeros of Q is neither of hyperelliptic type nor of abelian type, then allconclusions in the preceding paragraphs hold. Combining with Proposition 7.8 OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 49 (which joins ab-nonhyp to nonab) implies that Q has exactly one connectedcomponent.(2) Suppose the set of poles of Q is of both hyperelliptic and abelian types but not ofabelian-parity type. In this case the minimal stratum has three connected compo-nents, denoted by Q hypmin , Q abmin and Q nonhyp-nonabmin . Let C hyp , C ab and C nonhyp-nonab be the connected components of Q that are adjacent to the respective connectedcomponents of the minimal stratum.(a) If the set of zeros of Q is not of hyperelliptic type, we can choose an n j such that n j = P i = j n i . By Proposition 7.6 there is a path in Q ( n j , P i = j n i , − l , . . . , − l s )joining Q hypmin to a non-hyperelliptic component of Q min . Hence C hyp coincideswith one of the non-hyperelliptic components C ab and C nonhyp-nonab . It followsthat in this case the number of connected components of Q is at most two.(b) If the set of zeros of Q is not of abelian type, we can choose an n j such that n j is odd. By Proposition 7.8 there is a path in Q ( n j , P i = j n i , − l , . . . , − l s )joining Q abmin to Q nonabmin , which implies that C ab coincides with one of C hyp and C nonhyp-nonab . Hence in this case the number of connected components of Q isat most two.(c) If the set of zeros of Q is neither of hyperelliptic type nor of abelian type, then allconclusions in the preceding paragraphs hold. Combining with Proposition 7.8implies that Q has exactly one connected component.(3) Suppose the set of poles of Q is of abelian-parity type but not of hyperelliptictype. In this case the minimal stratum has three connected components, denotedby Q ab-evenmin , Q ab-oddmin and Q nonabmin . Let C ab-even , C ab-odd and C nonab be the connectedcomponents of Q that are adjacent to the respective connected components of theminimal stratum.(a) If the set of zeros of Q is of abelian type but not of abelian-parity type, wecan choose an n j not divisible by four. By Proposition 7.7 there is a pathin Q ( n j , P i = j n i , − l , . . . , − l s ) joining Q ab-evenmin to Q ab-oddmin . Hence C ab-even = C ab-odd , and in this case the number of connected components of Q is at mosttwo.(b) Suppose the set of zeros of Q is not of abelian type (hence not of abelian-parity type). We have seen that C ab-even = C ab-odd . Moreover, we can choosean odd n j . By Proposition 7.8 there is a path in Q ( n j , P i = j n i , − l , . . . , − l s )joining Q abmin to Q nonabmin , which implies that C nonab coincides with C ab-even = C ab-odd . Hence in this case Q has exactly one connected component.(4) Suppose the set of poles of Q is of hyperelliptic type but not of abelian type. Inthis case the minimal stratum has two connected components, denoted by Q hypmin and Q nonhypmin . Let C hyp and C nonhyp be the connected components of Q that areadjacent to the respective connected components of the minimal stratum.(a) If the set of zeros of Q is not of hyperelliptic type, we can choose an n j such that n j = P i = j n i . By Proposition 7.6 there is a path in Q ( n j , P i = j n i , − l , . . . , − l s )joining Q hypmin to Q nonhypmin . Hence C hyp coincides with C nonhyp . It follows that inthis case Q is connected. (5) Suppose the set of poles of Q is of abelian type but not of hyperelliptic typenor of abelian-parity type. In this case the minimal stratum has two connectedcomponents, denoted by Q abmin and Q nonabmin . Let C ab and C nonab be the connectedcomponents of Q that are adjacent to the respective connected components of theminimal stratum.(a) If the set of zeros of Q is not of abelian type, we can choose an odd n j . ByProposition 7.8 there is a path in Q ( n j , P i = j n i , − l , . . . , − l s ) joining Q abmin to Q nonabmin , which implies that C ab coincides with C nonab . It follows that in thiscase Q is connected.(6) For the remaining cases µ = (4 , , −
4) or (4 , , − , −
2) in g = 2, by Theorem 7.5(1b) the corresponding minimal strata have only three connected components, asthe hyperelliptic component coincides with one of the abelian-parity components.Therefore, the same argument as in (1a) implies that both strata have exactlythree connected components. In particular, we conclude that Q (4 , , − hyp = Q (4 , , − odd and Q (4 , , − , − hyp = Q (4 , , − , − even .Finally Theorem 7.1 follows from matching case by case with the above discussion,comparing the (same) upper and lower bounds on the number of connected componentsfor each case. (cid:3) Appendix A. Parity of k -differentials in genus zero and one Recall that for even k (and any genus) the parity of k -differentials of parity typecan be determined by using Remark 5.6. Nevertheless for odd k >
1, determiningthe parity explicitly can be challenging even for genus zero and one, despite that theclassification of connected components of the strata of k -differentials is known in theselow genus cases. Recall also that for odd k , a k -differential is of parity type if and onlyif the signature µ consists of even entries only (see Proposition 5.1).For g = 0, any k -differential ( X, ξ ) is determined up to scale by the positions of theunderlying singularities, which implies that Ω k M ( µ ) is a C ∗ -bundle over M ,n andhence is irreducible. For k = 1, since the canonical bundle of P has degree −
2, theparity of any (meromorphic) abelian differential with singularities of even order on P is always even. However for higher k , the canonical cover b X can have positive genus,hence determining the parity can be a non-trivial problem.For g = 1 and µ = (2 m , . . . , m n ), each connected component of Ω k M ( µ ) param-eterizes ( X, ξ ) such that the underlying divisor satisfies P ni =1 (2 m i /d ) z i ∼ X fora positive divisor d of gcd(2 m , . . . , m n ) (except d = 2 m for µ = (2 m, − m )) andsuch that any other torsion relation of the z i must be a multiple of this one for generic( X, ξ ) in the component. Recall that this number d is called the torsion number orequivalently the rotation number (see Proposition 3.13), and we denote by Ω k M ( µ ) d the corresponding component. For k = 1, the dimension h ( X, P ni =1 m i z i ) is either oneor zero, depending on whether P ni =1 m i z i ∼ k M ( µ ) d is odd if and only if d is even. However for higher k , similarly the canonical cover b X can have genus greater than or equal to two, hence determining the parity is again anon-trivial problem. OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 51 Our goal is thus to explicitly determine the parities of Ω k M ( µ ) and Ω k M ( µ ) d forodd k >
1. The strategy is to first study the base cases for k -differentials of genuszero with three singularities and k -differentials of genus one with two singularities, andthen apply induction to the number of singularities by smoothing certain multi-scale k -differentials.We begin with introducing several technical results which will be used in the subse-quent parity calculations.A.1. Technical tools.
Let (
X, ξ ) be a k -differential of type µ = (2 m , . . . , m n ). Re-call that for the canonical cover π : b X → X with π ∗ ξ = b ω k , there is a deck transfor-mation τ such that τ ∗ b ω = ζ b ω for a primitive k -th root of unity ζ . The singularities z , . . . , z n of ξ are the only (possible) branch points of π . Let r i = gcd( m i , k ), n i = m i /r i and ℓ i = k/r i . Then over z i there are r i distinct preimages x i, , . . . , x i,r i , each withmultiplicity ℓ i . The singularity order of b ω at each x i,j is 2 n i + ℓ i −
1. We denote by x i = P r i j =1 x i,j the reduced sum of the fiber points over z i .We first study certain τ -invariant divisors and their effective sections. Lemma A.1.
Let D be a τ -invariant divisor in b X . Then H ( b X, D ) has a basis given by(possibly meromorphic) functions f such that each underlying divisor ( f ) is τ -invariant.Moreover if deg D < k , then the support of the effective divisor D + ( f ) for any f inthis basis is a subset of { x i,j } for i = 1 , . . . , n and j = 1 , . . . , r i .Proof. Since τ k = Id and τ ∗ D = D , there is an eigenspace decomposition H ( b X, D ) = L ki =1 H ( b X, D ) i under the action of τ , where f ∈ H ( b X, D ) i satisfies that τ ∗ f = ζ i f .It implies that ( f ) = ( τ ∗ f ) = τ ∗ ( f ) give the same underlying divisor, hence thiseigenbasis provides a desired basis for H ( b X, D ).Next suppose deg
D < k . Then D + ( f ) is an effective τ -invariant divisor of degreesmaller than k . Suppose D + ( f ) = p + · · · + p j for j < k (where some p i can coincide).Then τ ∗ ( p + · · · + p j ) = p + · · · + p j . Note that τ permutes points in the same fiberof π , and away from the z i every fiber consists of k distinct points permuted cycliclyby τ . It implies that the p i must belong to the special fibers over the z i . (cid:3) Next we describe how to push down certain linear equivalence relations from b X to X . Lemma A.2. If P ni =1 a i x i ∼ holds in b X , then P ni =1 ( a i r i ) z i ∼ holds in X .Proof. By assumption, there exists a meromorphic function f on b X such that thedivisor of f is ( f ) = P ni =1 a i x i . Let h = Q kj =1 ( τ j ) ∗ f . Then h is τ -invariant, hence itcan also be regarded as a function on X . Note that z i = Q r i j =1 x ℓ i i,j under π (where weabuse notation to use the same symbol for both a point and a suitable local coordinate).Hence the factor Q r i j =1 x a i ki,j in h corresponds to z a i r i i . It follows that the divisor ( h )in X is equal to P ni =1 ( a i r i ) z i . (cid:3) Similarly we describe certain linear equivalence relations between the x i for thecanonical cover of a k -differential of genus zero. Lemma A.3.
Let ( X, ξ ) be a k -differential in the stratum Ω k M (2 m , . . . , m n ) ofgenus zero. Then the linear equivalence relations ℓ x ∼ · · · ∼ ℓ n x n ∼ − P ni =1 n i x i hold in the canonical cover b X .Proof. The relation ℓ i x i ∼ ℓ j x j follows from pulling back z i ∼ z j from X ∼ = P tothe canonical cover. To see the last relation, we give an explicit construction of thecanonical cover. Consider the cyclic cover π : b X → X modeled on x k = ( z − z ) m + k n Y i =2 ( z − z i ) m i (A.1)that maps ( x, z ) to z (after normalization if necessary to make b X smooth), where z is the (affine) coordinate of X ∼ = P and we assume that the z i are away from ∞ .Since P ni =1 m i = − k , the exponents on the right-hand side of Equation (A.1) sum tozero, hence the map is unramified over ∞ . Note that gcd( m + k, k ) = gcd( m , k ), themap has the correct ramification profile. To justify that it gives the canonical cover, itsuffices to show that P ni =1 (2 n i + ℓ i − x i ∼ K b X , which would then give the canonicaldivisor associated to an abelian differential as a k -th root of π ∗ ξ . To see this, notethat by Riemann-Hurwitz K b X ∼ π ∗ K X + P ni =1 ( ℓ i − x i , hence it reduces to show that π ∗ K X ∼ P ni =1 n i x i . Since X ∼ = P , its canonical divisor class can be represented by K X ∼ − z , and pulling it back via π gives π ∗ K X ∼ − ℓ x . Moreover, the associateddivisor of x (as a meromorphic function on b X ) is ( n + ℓ ) x + P ni =2 n i x i ∼
0. Therefore,we conclude that P ni =1 n i x i ∼ − ℓ x and ( b X, π ) is the desired canonical cover. (cid:3)
Remark A.4.
In Equation (A.1) one can replace the exponents of the ( z − z i ) by any a , . . . , a n such that a i ≡ m i (mod k ) for all i . Then by the same argument it stillgives the canonical cover up to isomorphism. The divisor ( x ) will differ from the aboveby adding some ℓ i x i and subtracting some other ℓ j x j , which gives the same relation asabove since ℓ i x i ∼ ℓ j x j for any i and j .A.2. Parity of k -differentials of genus zero. Let ξ be a k -differential on X ∼ = P of type µ = (2 m , . . . , m n ) for k odd. If ξ is not primitive, then by definition itsparity is equal to the parity of a primitive root differential. Note that ξ is primitive ifand only if gcd( m , . . . , m n , k ) = 1, which is equivalent to gcd( m , . . . , m n ) = 1 since P ni =1 m i = − k .We first consider k -differentials of genus zero with three singularities. Let ( X, ξ )be a k -differential in the stratum Ω k M (2 m , m , m ) with m + m + m = − k for k odd. Recall the notation that gcd( m i , k ) = r i , k = r i ℓ i and m i = r i n i . In thecanonical cover b X , we have π ∗ ξ = b ω k with the underlying canonical divisor given by( b ω ) = P i =1 (2 n i + ℓ i − x i , where x i = P r i j =1 x i,j with π − ( z i ) = { x i,j } for i = 1 , , ξ ) = h ( b X, ( b ω ) /
2) (mod 2).We first treat an easy case when some m i is divisible by k . Proposition A.5.
If some m i is divisible by k , then the parity of Ω k M (2 m , m , m ) is even.Proof. As remarked before we can assume that the stratum is primitive. Suppose m is divisible by k . Since gcd( m , m , m ) = 1 and m + m + m = − k , it implies OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 53 that r = r = 1. Then the genus of the canonical cover b X is zero. In this casedeg( b ω ) / −
1, hence h ( b X, ( b ω ) /
2) = 0 is even. (cid:3)
Next we consider the case when m , m and m are relatively prime to k . In orderto describe the parity explicitly, we introduce the following definition. Definition A.6.
For k odd and m + m + m = − k , we define N k ( m , m , m ) to bethe number of integral tuples ( c , c , c ) such that c , c , c ≥ , c + c + c = ( k − / X i =1 c i x i ∼ X i =1 ( m i + ( k − / x i modulo the linear equivalence relations kx ∼ kx ∼ kx ∼ − P i =1 m i x i in b X . Proposition A.7.
Suppose m , m and m are relatively prime to k . Then the parityof Ω k M (2 m , m , m ) is equal to the parity of N k ( m , m , m ) .Proof. The linear equivalence relations kx ∼ kx ∼ kx ∼ − P i =1 m i x i are alreadygiven in Lemma A.3. We show that any other relations between the x i must be gener-ated by these. Suppose first there is a relation between x i and x j , say, ax ∼ ax for aninteger a not divisible by k . Combining with kx ∼ kx , we can assume that a | k and0 < a < k . Set z = 0 and z = ∞ in X ∼ = P . Then the canonical cover π correspondsto a meromorphic function f on b X such that ( f ) = kx − kx and f is totally branchedat z (besides z and z ). Since ax − ax ∼
0, there is another function h on b X suchthat ( h ) = ax − ax . Therefore, up to scale f = h b , where b = k/a >
1. In otherwords, π factors through an intermediate cover b X → P → P where the first map is x y = h ( x ) and the second map is y z = y b . Since b >
1, this contradicts that π is totally branched at z , as the second map is only branched at 0 and ∞ .Next suppose there is a relation a x + a x ∼ ( a + a ) x . Since gcd( m i , k ) = 1,there exists w i such that m i w i ≡ k ). Hence multiplying the known relation m x + m x ∼ ( m + m ) x by a w leads to a x + a w m x ∼ a (1 + w m ) x .Combining these relations we conclude that ( a w m − a ) x ∼ ( a w m − a ) x . Bythe preceding paragraph, it implies that a ≡ a w m (mod k ), and hence the relation a x + a x ∼ ( a + a ) x is a multiple of the relation x + w m x ∼ (1 + w m ) x .This last relation follows from the known relation m x + m x ∼ ( m + m ) x bymultiplying by w (and subtracting the same amount of equivalent kx i on both sides).Finally we calculate h ( b X, ( b ω ) / b ω ) / k − / < k , by Lemma A.1the vector space H ( b X, ( b ω ) /
2) has a basis of meromorphic functions { f , . . . , f N } suchthat ( f i ) + P i =1 ( m i + ( k − / x i = P i =1 c i x i for c i ≥ c + c + c = ( k − / P i =1 c i x i is an effective divisor linearly equivalent to P i =1 ( m i + ( k − / x i .Moreover, two distinct such tuples ( c , c , c ) and ( c ′ , c ′ , c ′ ) must have c i = c ′ i for all i .Indeed if say c = c ′ , then we would have a relation ( c − c ′ ) x ∼ ( c − c ′ ) x for0 < | c − c ′ | < k , contradicting the established fact that it should be generated by therelation kx ∼ kx . It follows that the sections associated to such effective divisorshave mutually distinct zero or pole orders at x , hence they are linearly independent.In summary, we thus conclude that h ( b X, ( b ω ) /
2) = N k ( m , m , m ). (cid:3) Based on numerical evidence for small values of k (see Example A.12), we make thefollowing conjecture, which can be of independent interest in number theory. Conjecture A.8.
Suppose k is odd, m + m + m = − k and m , m , m are relativelyprime to k . Then N k ( m , m , m ) ≡ ⌊ k +14 ⌋ (mod 2) . Remark A.9.
Note that the relation m x + m x ∼ ( m + m ) x is equivalent to therelation x + nx ∼ (1 + n ) x , where n ≡ m w (mod k ) with m w ≡ k ). Itfollows that N k ( m , m , m ) = N k (1 , n, − k − − n ). Hence to prove the conjecture, itsuffices to consider the tuple (1 , n, − k − − n ) with gcd( n, k ) = gcd( n + 1 , k ) = 1.In the above let y i = x i − x be a divisor of degree zero for i = 1 ,
2. Then we have ky ∼ ky ∼ m y + m y ∼ b X . The condition P i =1 c i x i ∼ P i =1 ( m i + ( k − / x i and c , c , c ≥ k − / − c ) y + (( k − / − c ) y ∼ c , c ≥ c + c ≤ ( k − / c = ( k − / − c − c ≥ b i = ( k − / − c i for i = 1 ,
2. Combining with Remark A.9, we can formulate Conjecture A.8 in thefollowing equivalent form.
Conjecture A.10.
For k odd and gcd( n, k ) = gcd( n + 1 , k ) = 1 , let N k ( n ) be thenumber of pairs ( b , b ) such that b , b ≤ ( k − / , b + b ≥ ( k + 1) / and b ≡ nb (mod k ) . Then N k ( n ) ≡ ⌊ k +14 ⌋ (mod 2) . We provide some evidence and observation to this conjecture as follows.
Remark A.11.
We can directly verify Conjecture A.10 (and hence the equivalentConjecture A.8) for small values of k or n . For instance for n = 1, N k (1) is the numberof integers b such that ( k + 1) / ≤ b ≤ ( k − /
2. It is straightforward to check that N k (1) ≡ ⌊ k +14 ⌋ (mod 2).Moreover, let n ′ satisfy that nn ′ ≡ k ). Then ( n ′ + 1) n ≡ n (mod k ),hence gcd( n ′ , k ) = gcd( n ′ + 1 , k ) = 1. The condition b = nb (mod k ) is equivalent to b = n ′ b (mod k ), which implies that N k ( n ) = N k ( n ′ ). Example A.12.
We verify Conjecture A.10 (and hence the equivalent Conjecture A.8)for k up to 21. According to Remark A.11, in Table 2 we skip the case n = 1 andcombine the cases of n with its reciprocal n ′ (mod k ).Assuming Conjecture A.8 (or equivalently Conjecture A.10), we can indeed describethe parity of any k -differential in genus zero explicitly. We first introduce a parityfunction as follows. For k odd, consider the prime factorization k = p h · · · p h s s q ℓ · · · q ℓ t t where each p i is an odd prime such that ⌊ ( p i + 1) / ⌋ is even and each q i is an oddprime such that ⌊ ( q i + 1) / ⌋ is odd, namely, p i ≡ ± q i ≡ ± q and an integer m , denote by ν q ( m ) the q -adic evaluation given by the highestexponent ν such that q ν divides m . For the collection of primes q = { q , . . . , q t } inthe factorization of k , define ν q ( m ) = P ti =1 min { ν q i ( m ) , ν q i ( k ) } . Note that ν q ( k ) = P ti =1 ν q i ( k ) ≡ ⌊ ( k + 1) / ⌋ (mod 2).We can now introduce the number which will allows us to compute the parity ofstrata in genus zero. We remark that in Definition A.13 and Lemma A.14 below, wedo not require that µ is a partition of − k . OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 55 k n n ′ N k ( n ) ⌊ k +14 ⌋ k n n ′ N k ( n ) ⌊ k +14 ⌋
17 4 13 2 417 5 7 2 417 8 15 4 417 10 12 2 417 11 14 2 419 2 10 1 519 3 13 3 519 4 5 3 519 6 16 1 519 7 11 3 519 8 12 1 519 9 17 5 519 14 15 3 521 4 16 1 521 10 19 5 521 13 13 3 5
Table 2.
Verification of Conjecture A.10 for small values of k , where nn ′ ≡ k ). The last two columns have the same parity as pre-dicted by the conjecture. Definition A.13.
For k odd and µ = ( m , . . . , m n ), define n k ( µ ) to be the number ofentries m i in µ such that ν q ( m i ) ν q ( k ) (mod 2).The following properties of n k ( µ ) will be useful for our applications. Lemma A.14.
Let k = p h · · · p h s s q ℓ · · · q ℓ t t as above and µ = ( m , . . . , m n ) . (1) If ℓ i = 0 for all i , then n k ( µ ) ≡ for all µ . (2) The parity of n k ( µ ) only depends on the remainders of m , . . . , m n (mod k ) . (3) For µ = ( m , m , m ) such that m , m and m are relatively prime to k , we have n k ( µ ) ≡ ν q ( k ) (mod 2) . (4) If k and all entries of µ are divisible by d , then n k/d ( µ/d ) ≡ n k ( µ ) (mod 2) . (5) n k ( µ ) + n k ( µ ) ≡ n k ( µ , µ ) (mod 2) . (6) n k ( ± ℓ, ℓ, µ ) ≡ n k ( µ ) (mod 2) .Proof. If ℓ i = 0 for all i , then ν q ( m i ) = ν q ( k ) = 0 for all m i . Hence n k ( µ ) = 0,thus verifying item (1). Item (2) follows from the fact that min { ν q i ( m ) , ν q i ( k ) } =min { ν q i ( m + k ) , ν q i ( k ) } . For item (3), if m , m and m are relatively prime to k ,then ν q ( m i ) = 0 for all i . Hence n k ( µ ) = 0 if ν q ( k ) is even and n k ( µ ) = 3 if ν q ( k ) isodd. Item (4) follows from the fact that multiplying d to m i /d and to k/d changesthe respective ν q by the same amount. For items (5) and (6), they follow directly fromDefinition A.13. (cid:3) We first consider when k is an odd prime. In this case, if ⌊ ( k + 1) / ⌋ is odd, i.e. if k = q , then by definition n k ( µ ) is the number of entries in µ not divisible by k . If ⌊ ( k + 1) / ⌋ is even, i.e. if k = p , then n k ( µ ) is even by Lemma A.14 (1). Theorem A.15 (Conditional to Conjecture A.8) . Suppose k is an odd prime. Thenthe parity Φ(2 µ ) of Ω k M (2 µ ) is equal to n k ( µ ) (mod 2) .Proof. First suppose ⌊ ( k + 1) / ⌋ is odd. For µ = ( m , m , m ), either non of the m i is divisible by k , or exactly one of them is divisible by k , or all of them are divisibleby k . In the first case since k is prime, all m i are relatively prime to k , hence assumingConjecture A.8 the parity is odd, which coincides with the parity of n k ( µ ) = 3. In thesecond case by Proposition A.5 the parity is even, which coincides with the parity of n k ( µ ) = 2. In the last case the k -differential is a k -th power of an abelian differential,hence the parity is even, which coincides with the parity of n k ( µ ) = 0.Next we apply induction to the number of entries of µ . Suppose the claim holds forany µ with at most n − µ = ( m , . . . , m n ), if all the entries are divisibleby k , then the parity is even and n k ( µ ) = 0. Otherwise, there must be at least twoentries, say m and m , that are not divisible by k . By Proposition 3.1 and Lemma 5.7we can break up a zero of order 2 m + 2 m in a (meromorphic) k -differential of genuszero into two zeros of order 2 m and 2 m respectively and obtain the following parityrelationΦ(2 µ ) ≡ Φ(2 m , m , − k − m − m ) + Φ(2 m + 2 m , m , . . . , m n ) (mod 2) . If m + m is not divisible by k , then Φ(2 µ ) ≡ n k ( µ ) − ≡ n k ( µ ) (mod 2) holdsby induction. If m + m is divisible by k , then we obtain by induction the relationΦ(2 µ ) ≡ n k ( µ ) − ≡ n k ( µ ) (mod 2).Next suppose ⌊ ( k + 1) / ⌋ is even. Again for µ = ( m , m , m ), either non of the m i isdivisible by k , or exactly one of them is divisible by k , or all of them are divisible by k .The corresponding parities are all even (assuming Conjecture A.8 for the first case).Next we apply induction to the number of entries of µ . Suppose the claim holds for any µ with at most n − µ = ( m , . . . , m n ), if all the entries are divisible by k , then the parity is even. Otherwise, there must be at least two entries, say m and m , that are not divisible by k . By Proposition 3.1, Lemma 5.7 and using inductionwe conclude thatΦ(2 µ ) ≡ Φ(2 m , m , − k − m − m ) + Φ(2 m + 2 m , m , . . . , m n ) ≡ . (cid:3) Now we can show that n k ( µ ) is the desired parity function for general k (assumingConjecture A.8). Theorem A.16 (Conditional to Conjecture A.8) . The parity of Ω k M (2 µ ) is givenby n k ( µ ) (mod 2) .Proof. We apply induction to the exponents in the prime factorization of k . The claimholds for the base case when k is an odd prime by Theorem A.15. Suppose it holds forany exponents smaller than ( h , . . . , h s , ℓ , . . . , ℓ t ) (in the sense of multi-indices). OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 57 Consider first the case of three singularities µ = ( m , m , m ). If gcd( m , m , m ) = d >
1, then passing to a canonical d -cover and using induction, we have by Lemma A.14(4) that Φ k (2 µ ) ≡ Φ k/d (2 µ/d ) ≡ n k/d ( µ/d ) ≡ n k ( µ ) (mod 2), thus verifying the claimin this case.Next suppose gcd( m , m , m ) = 1. If m , m and m are relatively prime to k ,then assuming Conjecture A.8, Φ k (2 µ ) ≡ ⌊ ( k + 1) / ⌋ ≡ ν q ( k ) ≡ n k ( µ ) (mod 2) byLemma A.14 (3). Otherwise, there must be a prime factor d of k that divides one ofthe m i but not the other two, say, d divides m but not m and m . Then we can passto a canonical d -cover, which implies thatΦ k (2 µ ) = Φ k/d (2 m /d, . . . , m /d | {z } d , m + k − k/d, m + k − k/d ) . Using induction combined with Lemma A.14, the parity of this stratum is equal to n k/d (2 m /d, m , m ) ≡ n k (2 m , m d, m d ) (mod 2). Hence it suffices to showthat n k (2 m , m d, m d ) ≡ n k (2 m , m , m ) (mod 2), which is equivalent to showthat n k (2 m d, m d ) ≡ n k (2 m , m ) (mod 2). If d is equal to some prime p i , then n k (2 m p i , m p i ) = n k (2 m , m ) as p i is irrelevant to define n k in Definition A.13.If d is equal to some prime q i , then since m and m are not divisible by q i , it impliesthat ν q ( m q i ) = ν q ( m ) + 1 and ν q ( m q i ) = ν q ( m ) + 1. Therefore, the differencebetween n k (2 m q i , m q i ) and n k (2 m , m ) is even, thus verifying that they have thesame parity.Now we can apply another round of induction to the number of singularities. Sup-pose the claim holds for µ with at most n − n >
3. Then for µ = ( m , . . . , m n ) with P ni =1 m i = − k , combining Proposition 3.1, Lemma 5.7 andLemma A.14 we thus conclude thatΦ k (2 µ ) ≡ Φ k ( − k − m − m , m , m ) + Φ k (2 m + 2 m , m , . . . , m n ) ≡ n k ( m + m , m , m ) + n k ( m + m , m , . . . , m n ) ≡ n k ( m + m , m + m , m , m , m , . . . , m n ) ≡ n k ( m , . . . , m n ) (mod 2) . (cid:3) A.3.
Parity of k -differentials of genus one. We first consider k -differentials ingenus one with two singularities. Recall from Section 3.4 that the connected componentΩ k M (2 m, − m ) d parameterizes k -differentials of torsion number d , where d is a divisorof 2 m and d = 2 m . Theorem A.17.
For odd k , the parity of the component Ω k M (2 m, − m ) d is givenby d + 1 (mod 2) .Proof. Let (
X, ξ ) be a k -differential in the connected component Ω k M (2 m, − m ) d .Since the torsion number of ξ is d , we know that (2 m/d )( z − z ) ∼ X and norelation of lower order holds. Recall the notation gcd( m, k ) = r , k = rℓ and m = rn .In the canonical cover b X , we have π ∗ ξ = b ω k with the underlying canonical divisor( b ω ) = (2 n + ℓ − x + ( − n + ℓ − x being τ -invariant, where x i = P rj =1 x i,j with π − ( z i ) = { x i,j } rj =1 for i = 1 ,
2. Our goal is to evaluate Φ( ξ ) = h ( b X, ( b ω ) /
2) (mod 2).
Since deg( b ω ) / ℓ − < k , Lemma A.1 implies that H ( b X, ( b ω ) /
2) has a basis { f , . . . , f N } such that ( f i ) + ( n + ( ℓ − / x + ( − n + ( ℓ − / x ) = c i, x + c i, x for c i, , c i, ≥ c i, + c i, = ℓ −
1. Note that the Riemann-Hurwitz formula givesthe linear equivalence relation K b X ∼ ( ℓ − x + x ), hence ( b ω ) ∼ ( ℓ − x + x ) and2 nx ∼ nx . Therefore, the divisor class( n + ( ℓ − / x + ( − n + ( ℓ − / x ∼ ( n + ( ℓ − / x + ( − n + ( ℓ − / x is invariant when interchanging x and x . It implies that c i, x + c i, x is an effectivesection in the basis if and only if ( ℓ − − c i, ) x + ( ℓ − − c i, x ) is. Therefore, thedimension h ( b X, ( b ω ) /
2) is an odd number if and only if the linear equivalence relation( ℓ − x / ℓ − x / ∼ ( n + ( ℓ − / x + ( − n + ( ℓ − / x holds, i.e., if andonly if nx ∼ nx .Next we show that nx ∼ nx if and only if d is even. If nx ∼ nx , then byLemma A.2 we have mz ∼ mz , hence 2 m/d divides m by the assumption on therotation number, which is equivalent to d being even. Conversely if d is even, then mz ∼ mz , which implies that mℓx ∼ mℓx by pulling back via π . Note that wehave shown that 2 nx ∼ nx in the preceding paragraph. Since m = nr and ℓ, r areodd, we have gcd(2 n, mℓ ) = n . Combining the two linear equivalence relations of x and x , we thus conclude that nx ∼ nx . (cid:3) Theorem A.17 implies the following result which was previously used in Section 5.
Corollary A.18.
For k odd and every m ≥ , there exist cubic differentials in theprimitive locus Ω k M (2 m, − m ) prim with distinct parities.Proof. A connected component Ω k M (2 m, − m ) d with torsion number d parameterizesprimitive k -differentials if and only if gcd( k, d ) = 1. Since m ≥
2, we can choose d to be 1 and 2, both relatively prime to k . Then primitive k -differentials in the twoconnected components with torsion number one and two respectively have distinctparities according to Theorem A.17. (cid:3) Next we apply induction to the number of singularities and show that the questionreduces to determine the parity of k -differentials of genus zero. Proposition A.19.
Let µ = (2 m , . . . , m n ) be a signature of k -differentials in genusone. Let d be a common divisor of entries of µ such that d = ± m n . Then for k oddand n ≥ , the parity of the connected component Ω k M (2 m , . . . , m n ) d is equal to thesum of the parities of the connected component Ω k M (2 m n , − m n ) d and the stratum Ω k M (2 m , . . . , m n − , m n − k ) .Proof. Let ξ be a k -differential of genus one in Ω k M (2 m n , − m n ) d (which existsby the assumption that d = ± m n ). Let ξ be a k -differential of genus zero in theconnected stratum Ω k M (2 m , . . . , m n − , m n − k ). We can construct a multi-scale k -differential by gluing the singularity of ξ with order − m n to the singularity of ξ with order 2 m n − k . Then the claim follows from Lemma 5.7. (cid:3) If we choose d = 1 in Proposition A.19, then there is no restriction on m n , andobviously we can also use any other m i instead of m n . It thus implies the followingrelation for the strata of k -differentials in genus zero. OMPONENTS OF THE STRATA OF k -DIFFERENTIALS 59 Corollary A.20.
The parity of the strata Ω k M (2 m , . . . , m n ) in genus zero onlydepends on the remainders of m , . . . , m n (mod k ) . Note that Corollary A.20 coincides with our expectation in Lemma A.14 (2).Finally using the (conjectural) parity description of k -differentials in genus zero, wecan determine the parity of k -differentials in genus one. Recall the function n k ( µ )introduced in Definition A.13. Theorem A.21 (Conditional to Conjecture A.8) . The parity of the connected compo-nent Ω k M (2 µ ) d is given by n k ( µ ) + d + 1 (mod 2) .Proof. For µ = ( m , . . . , m n ), if there exists some m i such that d = ± m i , then theclaim follows from combining Proposition A.19, Theorem A.16 and Theorem A.17.If d = 2 | m i | for all i , then µ = ( m, . . . , m | {z } h , − m, . . . , − m | {z } h ) for some h ≥ d = 2 m .By a similar argument as in the proof of Proposition A.19, the parity of Ω k M (2 µ ) m is equal to the sum of the parities of the connected component Ω k M (2 m, m, − m ) m and the connected stratum Ω k M (2 m, . . . , m | {z } h − , − m, . . . , − m | {z } h , m − k ). The formerhas parity given by n k ( m, m, − m ) + 1 as shown in the preceding paragraph. Thelatter has parity given by n k ( m, . . . , m | {z } h − , − m, . . . , − m | {z } h , m ) using Theorem A.16 andLemma A.14 (2). Finally by Lemma A.14 (5) and (6), their sum has parity equalto the parity of n k ( µ ) + 1, thus completing the proof. Indeed in this special casesince there are even number of entries all of which have the same absolute value, byLemma A.14 (6) the parity of n k ( µ ) is even. (cid:3) Remark A.22.
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Department of Mathematics, Boston College, Chestnut Hill, MA 02467, USA
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