Hyperelliptic classes are rigid and extremal in genus two
aa r X i v : . [ m a t h . AG ] O c t HYPERELLIPTIC CLASSES ARE RIGID AND EXTREMAL IN GENUS TWO
VANCE BLANKERSA bstract . We show that the class of the locus of hyperelliptic curves with ℓ marked Weierstrasspoints, m marked conjugate pairs of points, and n free marked points is rigid and extremal in thecone of effective codimension-( ℓ + m ) classes on M + + n . This generalizes work of Chen andTarasca and establishes an infinite family of rigid and extremal classes in arbitrarily-high codimen-sion. I ntroduction Every smooth curve of genus two admits a unique degree-two hyperelliptic map to P . TheRiemann-Hurwitz formula forces such a map to have six ramification points called Weierstrasspoints ; each non-Weierstrass point p exists as part of a conjugate pair ( p, p ′ ) such that the imagesof p and p ′ agree under the hyperelliptic map.The locus of curves of genus two with ℓ marked Weierstrass points is codimension ℓ insidethe moduli space M , and in [CT16] it is shown that the class of the closure of this locus is rigidand extremal in the cone of effective classes of codimension ℓ . Our main theorem extends theirresult to H ⊆ M + + n , the locus of genus-two curves with ℓ marked Weierstrass points, m marked conjugate pairs, and n free marked points (see Definition 2.1). Main Theorem.
For ℓ, m, n ≥ , the class H , if non-empty, is rigid and extremal in the cone ofeffective classes of codimension ℓ + m in M + + n . In [CC15], the authors show that the effective cone of codimension-two classes of M hasinfinitely many extremal cycles for every n . Here we pursue a perpendicular conclusion: al-though in genus two ℓ ≤ , the number of conjugate pairs and number of free marked points areunbounded, so that the classes H form an infinite family of rigid and extremal cycles inarbitrarily-high codimension. Moreover, the induction technique used to prove the main result isgenus-agnostic, pointing towards a natural extension of the main theorem to higher genus givena small handful of low-codimension cases.When ℓ + m ≥ , our induction argument (Theorem 2.4) is a generalization of that used in[CT16, Theorem 4] to include conjugate pairs and free points; it relies on pushing forward aneffective decomposition of one hyperelliptic class onto other hyperelliptic classes and showingthat the only term of the decomposition to survive all pushforwards is the original class itself.This process is straightforward when there are at least three codimension-one conditions avail-able to forget; however, when ℓ + m = , and in particular when ℓ = and m = , more caremust be taken. The technique used in [CT16, Theorem 5] to overcome this problematic subcase The author was supported by NSF FRG grant 1159964 (PI: Renzo Cavalieri). elies on an explicit expression for (cid:2) H (cid:3) which becomes cumbersome when a non-zero num-ber of free marked points are allowed. Although adding free marked points can be describedvia pullback, pullback does not preserve rigidity and extremality in general, so we introduce anintersection-theoretic calculation using tautological ω -classes to handle this case instead.The base case of the induction (Theorem 2.2) is shown via a criterion (Lemma 1.4) given by[CC14] for rigidity and extremality for divisors; it amounts to an additional pair of intersectioncalculations. We utilize the theory of moduli spaces of admissible covers to construct a suitablecurve class for the latter intersection, a technique which generalizes that used in [Rul01] for theclass of H . Structure of the paper.
We begin in §1 with some background on M g,n and cones of effectivecycles. This section also contains the important Lemma 1.4 upon which Theorem 2.2 depends. In§2, we prove Theorem 2.2, which establishes the base case for the induction argument of our mainresult, Theorem 2.4. Finally, we conclude in §3 with a discussion of extending these techniquesfor g ≥ and possible connections to a CohFT-like structure. Acknowledgments.
The author wishes to thank Nicola Tarasca, who was kind enough to reviewan early version of the proof of the main theorem and offer his advice. The author is also greatlyindebted to Renzo Cavalieri for his direction and support.1. P reliminaries on M g,n and effective cycles Moduli spaces of curves, hyperelliptic curves, and admissible covers.
We work throughout in M g,n , the moduli space of isomorphism classes of stable genus g curves with n (ordered) markedpoints. If − + n > 0 this space is a smooth Deligne-Mumford stack of dimension − + n .We denote points of M g,n by [ C ; p , . . . , p n ] with p , . . . , p n ∈ C smooth marked points. Forfixed g , we may vary n to obtain a family of moduli spaces related by forgetful morphisms : foreach ≤ i ≤ n , the map π p i : M g,n → M g,n − forgets the i th marked point and stabilizes thecurve if necessary. The maps ρ p i : M g,n → M g, { p i } are the rememberful morphisms which are thecomposition of all possible forgetful morphisms other than π p i .Due to the complexity of the full Chow ring of M g,n , the tautological ring R ∗ ( M g,n ) is oftenconsidered instead [FP05] (for both rings we assume rational coefficients). Among other classes,this ring contains the classes of the boundary strata, as well as all ψ - and λ -classes. For ≤ i ≤ n the class ψ p i is defined to be the first Chern class of the line bundle on M g,n whose fiber overa given isomorphism class of curves is the cotangent line bundle at the i th marked point ofthe curve; λ is the first Chern class of the Hodge bundle. The tautological ring also includespullbacks of all ψ - and λ -classes, including the ω -classes , sometimes called stable ψ -classes . Theclass ω p i is defined on M g,n for g, n ≥ as the pullback of ψ p i along ρ p i . Several other notablecycles are known to be tautological, including the hyperelliptic classes considered below ([FP05]). Hyperelliptic curves are those which admit a degree-two map to P . The Riemann-Hurwitzformula implies that a hyperelliptic curve of genus g contains + Weierstrass points whichramify over the branch locus in P . For a fixed genus, specifying the branch locus allows one torecover the complex structure of the hyperelliptic curve and hence the hyperelliptic map. Thusfor g ≥ , the codimension of the locus of hyperelliptic curves in M g,n is g − . In this context, p p p p p +− p p p p p p p p p w1p p p p p +− F igure
1. On the left-hand side, the topological pictures of the general elementsof W (top) and γ (bottom) in M with P = { p , p , p } . On the right-handside, the corresponding dual graphs.requiring that a marked point be Weierstrass (resp. two marked points be a conjugate pair) is acodimension-one condition for genus at least two.We briefly use the theory of moduli spaces of admissible covers to construct a curve in M inTheorem 2.2. These spaces are particularly nice compactifications of Hurwitz schemes. For athorough introduction, the standard references are [HM82] and [ACV01]. For a more hands-onapproach in the same vein as our usage, see as well [Cav06]. Notation.
We use the following notation for boundary strata on M g,n ; all cycles classes aregiven as stack fundamental classes. For g ≥ , the divisor class of the closure of the locus ofirreducible nodal curves is denoted by δ irr . By δ h,P we mean the class of the divisor whosegeneral element has one component of genus h attached to another component of genus g − h ,with marked points P on the genus h component and marked points { p , . . . , p n }\ P on the other.By convention δ h,P = for unstable choices of h and P .Restrict now to the case of g = . We use W to denote the codimension-two class of thestratum whose general element agrees with that of δ , with the additional requirement thatthe node be a Weierstrass point. We denote by γ the class of the closure of the locus ofcurves whose general element has a genus component containing the marked points P meetingin two points conjugate under a hyperelliptic map a rational component with marked points { p , . . . , p n }\ P (see Figure 1).The space Adm − → ,...,t ,u ± ,...,u n ± is the moduli space of degree-two admissible covers ofgenus two with marked ramification points (Weierstrass points) t i and marked pairs of points(conjugate pairs) u j + and u j − . This space comes with a finite map c to M { t ,...,t ,u ,...,u n } whichforgets the cover and remembers only the base curve and its marked points, which are the images t t t t t t u + u − t t t t t t u F igure
2. An admissible cover in
Adm − → ,...,t ,u ± represented via dual graphs.In degree two the topological type of the cover is uniquely recoverable from thedual graph presentation.of the markings on the source. It comes also with a degree n map s to M + n which forgets thebase curve and all u j + and t i other than t and remembers the (stabilization of the) cover. ω -class lemmas. The following two lemmas concerning basic properties of ω -classes prove use-ful in the last subcase of Theorem 2.4. The first is a unique feature of these classes, and thesecond is the ω -class version of the dilaton equation. Lemma 1.1.
Let g ≥ , n ≥ , and P ⊂ { p , . . . , p n } such that | P | ≤ n − . Then for any p i , p j Pω p i · δ g,P = ω p j · δ g,P on M g,n .Proof. This follows immediately from Lemma 1.9 in [BC18]. (cid:3)
Lemma 1.2.
Let g, n ≥ . Then on M g,n , π p i ∗ ω p j = − if i = j , and otherwise.Proof. Let P = { p , . . . , p n } . When i = j , the pushforward reduces to the usual dilaton equationfor ψ p i on M g, { p i } . If π is the morphism which forgets all marked points, the diagram M g,n M g,P \{ p i } M g, { p i } M gρ pi π pi ππ pi commutes, so π p i ∗ ω p i = π p i ∗ ρ ∗ p i ψ p i = π ∗ π p i ∗ ψ p i = ( − ) .If i = j , then π p i ∗ ω p j = π p i ∗ π ∗ p i ω p j = . (cid:3) ones and properties of effective classes. For a projective variety X , the sum of two effectivecodimension- d classes is again effective, as is any Q + -multiple of the same. This gives a naturalconvex cone structure on the set of effective classes of codimension d inside the Q vector spaceof all codimension- d classes, called the effective cone of codimension- d classes and denoted Eff d ( X ) .Given an effective class E in the Chow ring of X , an effective decomposition of E is an equality E = m X s = a s E s with a s > 0 and E s irreducible effective cycles on X for all s . The main properties we are interestedin for classes in the pseudo-effective cone are rigidity and extremality. Definition 1.3.
Let E ∈ Eff d ( X ) . E is rigid if any effective cycle with class rE is supported on the support of E . E is extremal if, for any effective decomposition of E , all E s are proportional to E .When d = , elements of the cone correspond to divisor classes, and the study of Eff ( M g,n ) isfundamental in the theory of the birational geometry of these moduli spaces. For example, M is known to fail to be a Mori dream space for n ≥ (first for n ≥ in [CT15], then for n ≥ in [GK16], and the most recent bound in [HKL16]). For n ≥ in genus one, [CC14] show that M is not a Mori dream space; the same statement is true for M by [Mul17]. In these andselect other cases, the pseudo-effective cone of divisors has been shown to have infinitely manyextremal cycles and thus is not rational polyhedral ([CC15]).These results are possible due in large part to the following lemma, which plays an importantrole in Theorem 2.2. Here a moving curve C in D is a curve C , the deformations of which cover aZariski-dense subset of D . Lemma 1.4 ([CC14, Lemma 4.1]) . Let D be an irreducible effective divisor in a projective variety X , andsuppose that C is a moving curve in D satisfying Z X [ D ] · [ C ] < 0 . Then [ D ] is rigid and extremal. (cid:3) Remark 1.5.
Using Lemma 1.4 to show a divisor D is rigid and extremal in fact shows more: ifthe lemma is satisfied, the boundary of the pseudo-effective cone is polygonal at D . We do notrely on this fact, but see [Opi16, §6] for further discussion.Lemma 1.4 allows us to change a question about the pseudo-effective cone into one of in-tersection theory and provides a powerful tool in the study of divisor classes. Unfortunately, itfails to generalize to higher-codimension classes, where entirely different techniques are needed.Consequently, much less is known about Eff d ( M g,n ) for d ≥ . This paper is in part inspiredby [CT16], where the authors show that certain hyperelliptic classes of higher codimension arerigid and extremal in genus two. In [CC15], the authors develop additional extremality criteriato show that in codimension-two there are infinitely many extremal cycles in M for all n ≥ and in M for all n ≥ , as well as showing that two additional hyperelliptic classes of highergenus are extremal. These criteria cannot be used directly for the hyperelliptic classes we con-sider; this is illustrative of the difficulty of proving rigidity and extremality results for classes ofcodimension greater than one. . M ain theorem In this section we prove our main result, which culminates in Theorem 2.4. The proof proceedsvia induction, with the base cases given in Theorem 2.2. We begin by defining hyperellipticclasses on M g,n . Definition 2.1.
Fix integers ℓ, m, n ≥ . Denote by H g,ℓ,2m,n the closure of the locus of hyper-elliptic curves in M g,ℓ + + n with marked Weierstrass points w , . . . , w ℓ ; pairs of marked points + , − , . . . , + m , − m with + j and − j conjugate under the hyperelliptic map; and free marked points p , . . . , p n with no additional constraints. By hyperelliptic class , we mean a non-empty class equiv-alent to some (cid:2) H g,ℓ,2m,n (cid:3) in the Chow ring of M g,ℓ + + n . w w p p p F igure
3. The general element of H .Lemma 1.4 allows us to establish the rigidity and extremality of the two divisor hyperellipticclasses for genus two, which together provide the base case for Theorem 2.4. Theorem 2.2.
For n ≥ , the class of H is rigid and extremal in Eff ( M + n ) and the class of H is rigid and extremal in Eff ( M + n ) .Proof. Define a moving curve C in H by fixing a general genus-two curve C with n freemarked points p , . . . , p n and varying the conjugate pair (+ , −) .Since (cid:2) H (cid:3) = π ∗ p n · · · π ∗ p (cid:2) H (cid:3) , by the projection formula and the identity (see [Log03]) (cid:2) H (cid:3) = − λ + ψ + + ψ − − ∅ − δ ∅ , we compute Z M + n (cid:2) H (cid:3) · [ C ] = Z M (cid:2) H (cid:3) · π p ∗ · · · π p n ∗ [ C ]= + ( − + ) + ( − + ) − ( ) − = − In particular, intersecting with λ is by projection formula. Intersecting with either ψ -class canbe seen as follows: pullback ψ i from M to ψ i − δ ∅ , then use projection formula on ψ i backto M . This is just − , since ψ i is the first Chern class of the cotangent bundle of C over i . The intersection with δ ∅ corresponds to the + Weierstrass points. Finally, δ ∅ intersectstrivially, since by fixing C we have only allowed rational tail degenerations.As (cid:2) H (cid:3) is irreducible, it is rigid and extremal by Lemma 1.4.We next apply Lemma 1.4 by constructing a moving curve B which intersects negatively with H using the following diagram. Note that the image of s is precisely H ⊂ M + n . dm − → ,...,t ,u ± ,...,u n ± M + n M { t ,...,t ,u ,...,u n } M { t ,...,t ,u ,...,u n } c sπ t6 Fix the point [ b n ] in M { t ,...,t ,u ,...,u n } corresponding to a chain of P s with n + componentsand marked points as shown in Figure 4 (if n = , t and t are on the final component; if n = , t and u are on the final component; etc.). Then [ B n ] = s ∗ c ∗ π ∗ t [ b n ] is a moving curve in H (after relabeling t to w and u j − to p j ). t t t t u n − u n − u n − u n F igure
4. The point [ b n ] in M { t ,...,t ,u ,...,u n } .The intersection (cid:2) H (cid:3) · [ B n ] is not transverse, so we correct with minus the Euler class ofthe normal bundle of H in M + n restricted to B n . In other words, Z M + n (cid:2) H (cid:3) · [ B n ] = Z M + n − π ∗ p n · · · π ∗ p ψ w · [ B n ]= Z M − ψ w · [ B ] . By passing to the space of admissible covers, this integral is seen to be a positive multiple (apower of two) of Z M − ψ w · (cid:2) H (cid:3) = Z M − ψ w · ( w )= −
18 , where we have used the fact that (cid:2) H (cid:3) = w [Cav16]. (cid:3) This establishes the base case for the inductive hypothesis in Theorem 2.4. The inductionprocedure differs fundamentally for the codimension-two classes, so we first prove the followingshort lemma to simplify the most complicated of those.
Lemma 2.3.
The class W { p ,...,p n } is not proportional to (cid:2) H (cid:3) on M + n .Proof. Let P = { p , . . . , p n } . Note that in W the marked points w and w carry no specialrestrictions, and the class is of codimension two. By dimensionality on the rational componentof the general element of W , W · ψ n + = owever, using the equality (cid:2) H (cid:3) = w ψ w − ( ψ + ψ ) − ( ψ w + ψ w ) (cid:18) { w } +
35 δ ∅ +
120 δ irr (cid:19) established in [CT16, Equation 4] and Faber’s Maple program [Fab], we compute Z M + n (cid:2) H (cid:3) · ψ n + = Z M + n π ∗ p · · · π ∗ p n (cid:2) H (cid:3) · ψ n + = Z M (cid:2) H (cid:3) · π p ∗ · · · π p n ∗ ψ n + = Z M w ψ w − ( ψ + ψ )− ( ψ w + ψ w ) (cid:18) { w } +
35 δ ∅ +
120 δ irr (cid:19) ! · ψ = so W is not a non-zero multiple of (cid:2) H (cid:3) . (cid:3) We are now ready to prove our main result. The bulk of the effort is in establishing extremality,though the induction process does require rigidity at every step as well. Although we do notinclude it until the end, the reader is free to interpret the rigidity argument as being applied ateach step of the induction.The overall strategy of the extremality portion of the proof is as follows. Suppose (cid:2) H (cid:3) isgiven an effective decomposition. We show (first for the classes of codimension at least three, thenfor those of codimension two) that any terms of this decomposition which survive pushforwardby π w i or π + j must be proportional to the hyperelliptic class itself. Therefore we may write (cid:2) H (cid:3) as an effective decomposition using only classes which vanish under pushforwardby the forgetful morphisms; this is a contradiction, since the hyperelliptic class itself survivespushforward. Theorem 2.4.
For ℓ, m, n ≥ , the class H , if non-empty, is rigid and extremal in Eff ℓ + m ( M + + n ) .Proof. We induct on codimension; assume the claim holds when the class is codimension ℓ + m − .Theorem 2.2 is the base case, so we may further assume ℓ + m ≥ . Now, suppose that (cid:2) H (cid:3) = X s a s [ X s ] + X t b t [ Y t ] (1)is an effective decomposition with [ X s ] and [ Y t ] irreducible codimension- ( ℓ + m ) effective cycleson M + + n , with [ X s ] surviving pushforward by some π w i or π + j and [ Y t ] vanishing under allsuch pushforwards, for each s and t .Fix an [ X s ] appearing in the right-hand side of (1). If ℓ = , suppose without loss of generality(on the w i ) that π w ∗ [ X s ] = . Since π w ∗ (cid:2) H (cid:3) = ( − ( ℓ − )) (cid:2) H − (cid:3) is rigid and extremal by hypothesis, π w ∗ [ X s ] is a positive multiple of the class of H − and X s ⊆ ( π w ) − H − . By the commutativity of the following diagrams and the observation hat hyperelliptic classes survive pushforward by all π w i and π + j , we have that π w i ∗ [ X s ] = and π + j ∗ [ X s ] = for all i and j . H H ( m − ) ,n + H H − H − H − ( m − ) ,n + H − H − w1 π + j π w1 π w1 π wi π w1 π + j π wi If ℓ = , suppose without loss of generality (on the + j ) that π + ∗ [ X s ] = . Then the sameconclusion holds that [ X s ] survives all pushforwards by π + j , since π + ∗ (cid:2) H (cid:3) = (cid:2) H ( m − ) ,n + (cid:3) is rigid and extremal by hypothesis, and π + commutes with π + j .It follows that for any ℓ + m ≥ s ⊆ \ i,j (cid:16) ( π w i ) − H − ∩ ( π + j ) − H ( m − ) ,n + (cid:17) . We now have two cases. If ℓ + m ≥ , any ℓ + − Weierstrass or conjugate pair marked pointsin a general element of X s are distinct, and hence all ℓ + such marked points in a generalelement of X s are distinct. We conclude that [ X s ] is a positive multiple of (cid:2) H (cid:3) . If ℓ + m = ,we must analyze three subcases.If ℓ = and m = , then X s ⊆ ( π + ) − H + ∩ ( π + ) − H + . The modular interpretation of the intersection leaves three candidates for [ X s ] : W or γ forsome P containing neither conjugate pair, or (cid:2) H (cid:3) itself. However, for the former two,dim W = dim π + ( W ) and dim γ = dim π + ( γ ) for all such P , contradicting our as-sumption that the class survived pushforward. Thus [ X s ] is proportional to (cid:2) H (cid:3) .If ℓ = and m = , similar to the previous case, [ X s ] could be (cid:2) H (cid:3) or W or γ forsome P containing neither the conjugate pair nor the Weierstrass point. However, if X s is eitherof the latter cases, we have dim X s = dim π + ( X s ) , again contradicting our assumption about thenon-vanishing of the pushforward, and so again [ X s ] must be proportional to (cid:2) H (cid:3) .If ℓ = and m = , as before, [ X s ] is either (cid:2) H (cid:3) itself or W or γ for P = { p , . . . , p n } .Now dim W = dim π w i W , so the argument given in the other subcases fails (though γ isstill ruled out as before). Nevertheless, we claim that W cannot appear on the right-hand sideof (1) for H ; to show this we induct on the number of free marked points n . The base caseof n = is established in [CT16, Theorem 5], so assume that H − is rigid and extremal forsome n ≥ . Suppose for the sake of contradiction that (cid:2) H (cid:3) = a W + X s a s [ Z s ] (2)is an effective decomposition with each [ Z s ] an irreducible codimension-two effective cycle on M + n . Note that W = π ∗ p n W \{ p n } − W \{ p n } . ultiply (2) by ω p n and push forward by π p n . On the left-hand side, π p n ∗ (cid:0) ω p n · (cid:2) H (cid:3)(cid:1) = π p n ∗ (cid:0) ω p n · π ∗ p n (cid:2) H − (cid:3)(cid:1) = π p n ∗ ( ω p n ) · (cid:2) H − (cid:3) = (cid:2) H − (cid:3) , having applied Lemma 1.2. Combining this with the right-hand side, (cid:2) H − (cid:3) = a π p n ∗ (cid:0) ω p n · π ∗ p n W \{ p n } − ω p n · W \{ p n } (cid:1) + X s a s π p n ∗ ( ω p n · [ Z s ])= W \{ p n } + π p n ∗ (cid:0) ω p n · W \{ p n } (cid:1) + X s a s π p n ∗ ( ω p n · [ Z s ]) . The term π p n ∗ (cid:0) ω p n · W \{ p n } (cid:1) vanishes by Lemma 1.1: π p n ∗ (cid:0) ω p n · W \{ p n − } (cid:1) = π p n ∗ (cid:0) ω w · W \{ p n } (cid:1) = π p n ∗ (cid:0) π ∗ p n ω w · W \{ p n } (cid:1) = ω w · π p n ∗ W \{ p n } = where w is the Weierstrass singular point on the genus two component of W \{ p n } . Altogether,we have (cid:2) H − (cid:3) = W \{ p n } + X s a s π p n ∗ ( ω p n · [ Z s ]) . [Rul01] establishes that ψ p n is semi-ample on M { p n } , so ω p n is semi-ample, and hence this isan effective decomposition. By hypothesis, H − is rigid and extremal, so W \{ p n } must bea non-zero multiple of (cid:2) H − (cid:3) , which contradicts Lemma 2.3. Therefore W cannot appearas an [ X s ] in (1).Thus for all cases of ℓ + m = (and hence for all ℓ + m ≥ ), we conclude that each [ X s ] in (1)is a positive multiple of (cid:2) H (cid:3) . Now subtract these [ X s ] from (1) and rescale, so that (cid:2) H (cid:3) = X t b t [ Y t ] . Recall that each [ Y t ] is required to vanish under all π w i ∗ and π + j ∗ . But the pushforward of (cid:2) H (cid:3) by any of these morphisms is non-zero, so there are no [ Y t ] in (1). Hence (cid:2) H (cid:3) isextremal in Eff ℓ + m ( M + + n ) .For rigidity, suppose that E := r (cid:2) H (cid:3) is effective. Since π w i ∗ E = ( − ( ℓ − )) r (cid:2) H − (cid:3) and π + j ∗ E = r (cid:2) H ( m − ) ,n + (cid:3) are rigid and extremal for all i and j , we have that π w i ∗ E is sup-ported on H − and π + j ∗ E is supported on H ( m − ) ,n + . This implies that E is supportedon the intersection of ( π w i ) − (cid:2) H − (cid:3) and ( π + j ) − (cid:2) H ( m − ) ,n + (cid:3) for all i and j . Thus E issupported on H , so (cid:2) H (cid:3) is rigid. (cid:3)
3. H igher genus
The general form of the inductive argument in Theorem 2.4 holds independent of genus for g ≥ . However, for genus greater than one, the locus of hyperelliptic curves in M g is ofcodimension g − , so that the base cases increase in codimension as g increases. The challenge in howing the veracity of the claim for hyperelliptic classes in arbitrary genus is therefore wrappedup in establishing the base cases of codimension g − (corresponding to Theorem 2.2) andcodimension g (corresponding to the three ℓ + m = subcases in Theorem 2.4).In particular, our proof of Theorem 2.2 relies on the fact that H and H are divisors,and the subcase ℓ = in Theorem 2.4 depends on our ability to prove Lemma 2.3. This in turnrequires the description of H given by [CT16]. More subtly, we require that ψ p n be semi-ample in M { p n } , which is known to be false in genus greater than two in characteristic 0 [Kee99].In genus three, [CC15] show that the base case H is rigid and extremal, though it is unclearif their method will extend to H . Moreover, little work has been done to establish the caseof a single conjugate pair in genus three, and as the cycles move farther from divisorial classes,such analysis becomes increasingly more difficult.One potential avenue to overcome these difficulties is suggested by work of Renzo Cavalieriand Nicola Tarasca (currently in preparation). They use an inductive process to describe hy-perelliptic classes in terms of decorated graphs using the usual dual graph description of thetautological ring of M g,n . Such a formula for the three necessary base cases would allow forgreatly simplified intersection-theoretic calculations, similar to those used in Theorem 2.2 andLemma 2.3. Though such a result would be insufficient to completely generalize our main theo-rem, it would be a promising start.We also believe the observation that pushing forward and pulling back by forgetful morphismsmoves hyperelliptic classes to (multiples of) hyperelliptic classes is a useful one. There is evidencethat a more explicit connection between marked Weierstrass points, marked conjugate pairs, andthe usual gluing morphisms between moduli spaces of marked curves exists as well, thoughconcrete statements require a better understanding of higher genus hyperelliptic loci. Althoughit is known that hyperelliptic classes do not form a cohomological field theory over the full M g,n ,a deeper study of the relationship between these classes and the natural morphisms among themoduli spaces may indicate a CohFT-like structure, which in turn would shed light on graphformulas or other additional properties. R eferences [ACV01] Dan Abramovich, Alessio Corti, and Angelo Vistoli, Twisted bundles and admissible covers , Comm. in Algebra (2001), no. 8, 3547–3618.[BC18] Vance Blankers and Renzo Cavalieri, Intersections of ω classes in M g,n , Proceedings of Gökova Geometry–Topology Conference 2017 (2018), 37–52.[Cav06] Renzo Cavalieri, Hodge-type integrals on moduli spaces of admissible covers , Geom. Topol. Monogr. (2006),167–194.[Cav16] , Hurwitz theory and the double ramification cycle , Jpn. J. Math. (2016), no. 2, 305–331.[CC14] Dawei Chen and Izzet Coskun, Extremal effective divisors on M , Math. Ann. (2014), no. 3-4, 891–908.[CC15] , Extremal higher codimension cycles on moduli spaces of curves , Proc. London Math. Soc. (2015), no. 1,181–204.[CT15] Ana-Maria Castravet and Jenia Tevelev, M is not a Mori dream space , Duke Math. J. (2015), no. 8,3851–3878.[CT16] Dawei Chen and Nicola Tarasca, Extremality of loci of hyperelliptic curves with marked Weierstrass points , Algebra& Number Theory (2016), no. 1, 1935–1948.[Fab] Carel Faber, M aple program for computing hodge integrals , Available at http://math.stanford.edu/~vakil/programs . FP05] Carel Faber and Rahul Pandharipande,
Relative maps and tautological classes , J. Eur. Math. Soc. (JEMS) (2005),no. 1, 13–49.[GK16] José Luis González and Kalle Karu, Some non-finitely generated Cox rings , Compos. Math. (2016), no. 5,984–996.[HKL16] Juergen Hausen, Simon Keicher, and Antonio Laface,
On blowing up the weighted projective plane , ArXiv e-prints (2016).[HM82] Joe Harris and David Mumford,
On the Kodaira dimension of the moduli space of curves , Invent. Math. (1982),23–88.[Kee99] Seán Keel, Basepoint freeness for nef and big line bundles in positive characteristic , Ann. of Math. (1999), no. 1,253–286.[Log03] Adam Logan,
The Kodaira dimension of moduli spaces of curves with marked points , Amer. J. Math. (2003),no. 1, 105–138.[Mul17] Scott Mullane,
On the effective cone of M g,n , arXiv:1701.05893 (2017).[Opi16] Morgan Opie, Extremal divisors on moduli spaces of rational curves with marked points , Michican Math. J. (2016), no. 2, 251–285.[Rul01] William Frederick Rulla, The birational geometry of moduli space M(3) and moduli space M(2,1) , Ph.D. thesis,University of Texas at Austin, 2001.D epartment of M athematics , C olorado S tate U niversity , F ort C ollins , C olorado E-mail address : [email protected]@math.colostate.edu