Moduli of theta-characteristics via Nikulin surfaces
aa r X i v : . [ m a t h . AG ] O c t MODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES
GAVRIL FARKAS AND ALESSANDRO VERRA
The importance of the locus K g := { [ C ] ∈ M g : C lies on a K surface } has beenrecognized for some time. Fundamental results in the theory of algebraic curves like theBrill-Noether Theorem [Laz], or Green’s Conjecture for generic curves [Vo] have beenproved by specialization to a general point [ C ] ∈ K g . The variety K g viewed as a subva-riety of M g serves as an obstruction for effective divisors on M g to having small slope[FP] and thus plays a significant role in determining the cone of effective divisors on M g .The first aim of this paper is to show that at the level of the the Prym moduli space R g classifying ´etale double covers of curves of genus g , the locus of curves lying on a Nikulin K surfaces plays a similar role. The analogy is far-reaching: Nikulin surfacesfurnish an explicit unirational parametrization of R g in small genus, see Theorem 0.2,just like ordinary K surfaces do the same for M g ; numerous results involving curveson K surfaces have a Prym-Nikulin analogue, see Theorem 0.4, and even exceptions touniform statements concerning curves on K surfaces carry over in this analogy!Our other aim is to complete the birational classification of the moduli space S + g ofeven spin curves of genus g . It is known [F] that S + g is of general type when g ≥ . UsingNikulin surfaces we show that S + g is uniruled for g ≤ , see Theorem 0.7, which leaves S +8 as the only case missing from the classification. We prove the following: Theorem 0.1.
The Kodaira dimension of S +8 is equal to zero. Theorems 0.1 and 0.7 highlight the fact that the birational type of S + g is entirelygoverned by the world of K surfaces, in the sense that S + g is uniruled precisely whena general even spin curve of genus g moves on a special K surface. This is in contrastto M g which is known to be uniruled at least for g ≤ , whereas the general curve ofgenus g ≥ does not lie on a K surface.A Nikulin surface [Ni] is a K surface S endowed with a non-trivial double cover f : ˜ S → S with a branch divisor N := N + · · · + N consisting of disjoint smooth rational curves N i ⊂ S . Blowing down the ( − -curves E i := f − ( N i ) ⊂ ˜ S , one obtains a minimal K surface σ : ˜ S → Y , together with an involution ι ∈ Aut ( Y ) having fixed points corre-sponding to the images σ ( E i ) of the exceptional divisors. The class O S ( N ) is divisible by in Pic( S ) and we set e := O S ( N + · · · + N ) ∈ Pic( S ) . Assume that C ⊂ S is a smoothcurve of genus g such that C · N i = 0 for i = 1 , . . . , . We say that the triple ( S, e, O S ( C )) is a polarized Nikulin surface of genus g and denote by F N g the -dimensional moduli spaceof such objects. Over F N g we consider the P g -bundle P N g := n ( S, e, C ) : C ⊂ S is a smooth curve such that [ S, e, O S ( C )] ∈ F N g o , which comes equipped with two maps P N gp g ~ ~ ~~~~~~~~ χ g (cid:31) (cid:31) @@@@@@@@ F N g R g where p g ([ S, e, C ]) := [
S, e, O S ( C )] and χ g ([ S, e, C ]) := [
C, e C := e ⊗ O C ] . Since C · N = 0 ,it follows that e ⊗ C = O C . The ´etale double cover induced by e C is precisely the restriction f C := f | ˜ C : ˜ C → C, where ˜ C := f − ( C ) . Note that dim ( P N g ) = 11 + g and it is natural toask when is χ g dominant and induces a uniruled parametrization of R g . Theorem 0.2.
The general Prym curve [ C, e C ] ∈ R g lies on a Nikulin surface if and only if g ≤ and g = 6 , that is, the morphism χ g : P N g → R g is dominant precisely in this range. In contrast, the general Prym curve [ C, e C ] ∈ R lies on an Enriques surface [V1]but not on a Nikulin surface. Since P N g is a uniruled variety being a P g -bundle over F N g ,we derive from Theorem 0.2 the following immediate consequence: Corollary 0.3.
The Prym moduli space R g is uniruled for g ≤ . The discussion in Sections 2 and 3 implies the stronger result that F N g (and thus N g := Im ( χ g ) ) is unirational for g ≤ . It was known that R g is rational for g ≤ , see[Do2], [Ca], and unirational for g = 5 , , see [Do], [ILS], [V1], [V2]. Apart from the resultin genus which is new, the significance of Corollary 0.3 is that Nikulin surfaces providean explicit uniform parametrization of R g that works for all genera g ≤ .Before going into a more detailed explanation of our results on F N g , it is instructiveto recall Mukai’s work on the moduli space F g of polarized K surfaces of genus g :Mukai’s results [M1], [M2], [M3]: (1) A general curve [ C ] ∈ M g lies on a K surface if and only if g ≤ and g = 10 , thatis, the equality K g = M g holds precisely in this range. (2) M is birationally isomorphic to the tautological P -bundle P over the modulispace F of polarized K surfaces of genus . There is a commutative diagram M o o ∼ = q " " FFFFFFFF P p | | zzzzzzzz F with q − ([ C ]) = [ S, C ] , where S is the unique K surface containing a general [ C ] ∈ M . (3) The locus K is a divisor on M which has the following set-theoretic incarnation: K = (cid:8) [ C ] ∈ M : ∃ L ∈ W ( C ) such that µ ( L ) : Sym H ( C, L ) ≇ −→ H ( C, L ⊗ ) (cid:9) . (4) There exists a rational variety X ⊂ P with K X = O X ( − and dim( X ) = 5 , such thatthe general K surface of genus appears as a -dimensional linear section of X . Sucha realization is unique up to the action of Aut( X ) and one has birational isomorphisms: F ∼ = G (cid:0) P , P (cid:1) ss // Aut( X ) and K ∼ = G (cid:0) P , P (cid:1) ss // Aut( X ) . ODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES 3
To this list of well-known results, one could add the following statement from [FP]: (5)
The closure K of K inside M is an extremal point in the effective cone Eff( M ) ;its class K ≡ λ − δ − δ − δ − δ − δ −· · · ∈ Pic( M ) has minimal slope among alleffective divisors on M and provides a counterexample to the Slope Conjecture [HMo].Quite remarkably, each of the statements (1)-(5) has a precise Prym-Nikulin ana-logue. Theorem 0.2 is the analogue of (1). For the highest genus when the Prym-Nikulincondition is generic, the moduli space acquires a surprising Mori fibre space structure: Theorem 0.4.
The moduli space R is birationally isomorphic to the tautological P -bundle P N and there is a commutative diagram: R o o ∼ = χ AAAAAAAA P N p ~ ~ }}}}}}}} F N Furthermore, χ − ([ C, η ]) = [
S, C ] , where the unique Nikulin surface S containing C is given bythe base locus of the net of quadrics containing the Prym-canonical embedding φ K C ⊗ η : C → P . Just like in Mukai’s work, the genus next to maximal from the point of view ofPrym-Nikulin theory, behaves exotically.
Theorem 0.5.
The Prym-Nikulin locus N := Im( χ ) is a divisor on R which can be identifiedwith the ramification locus of the Prym map Pr : R → A : N = (cid:8) [ C, η ] ∈ R : µ ( K C ⊗ η ) : Sym H ( C, K C ⊗ η ) ≇ −→ H ( C, K ⊗ C ) (cid:9) . Observe that both divisors K and N share the same Koszul-theoretic description. Fur-thermore, they are both extremal points in their respective effective cones, cf. Proposition3.6. Is there a Prym analogue of the genus Mukai G -variety X := G /P ⊂ P ? Theanswer to this question is in the affirmative and we outline the construction of a Grass-mannian model for F N while referring to Section 3 for details.Set V := C and U := C and view P = P ( U ) as the space of planes inside P ( U ∨ ) .Let us choose a smooth quadric Q ⊂ P ( V ) . The quadratic line complex W Q ⊂ G (2 , V ) ⊂ P ( ∧ V ) consisting of tangent lines to Q is singular along the codimension subvariety V Q of lines contained in Q . One can identify V Q with the Veronese -fold ν (cid:0) P (cid:1) ⊂ P (cid:0) Sym ( U ) (cid:1) = P ( ∧ V ) = P . The projective tangent bundle P Q of Q , viewed as the blow-up of W Q along V Q , is en-dowed with a double cover branched along V Q and induced by the map P × P −→ P (cid:0) Sym ( U ) (cid:1) , ( H , H ) H + H . We show in Theorem 3.4 that codimension linear sections of W Q are Nikulin surfacesof genus with general moduli. Moreover there is a birational isomorphism F N ∼ = G (7 , ∧ V ) ss // Aut( Q ) . Taking codimension linear sections of W Q one obtains a similar realization of N , whichshould be viewed as the Prym counterpart of Mukai’s construction of K . G. FARKAS AND A. VERRA
The subvariety K g ⊂ M g is intrinsic in moduli , that is, its generic point [ C ] admitscharacterizations that involve C alone and the K surface containing C is a result of somepeculiarity of the canonical curve. For instance [BM], if [ C ] ∈ K g then the Wahl map ψ K C : ∧ H ( C, K C ) → H ( C, K ⊗ C ) , is not surjective. It is natural to ask for similar intrinsic characterizations of the Prym-Nikulin locus N g ⊂ R g in terms of Prym curves alone, without making reference toNikulin surfaces. In this direction, we prove in Section 1 the following result: Theorem 0.6.
Set g := 2 i + 6 . Then K i, ( C, K C ⊗ η ) = 0 for any [ C, η ] ∈ N g , that is, thePrym-canonical curve C | K C ⊗ η | −→ P g − of a Prym-Nikulin section fails to satisfy property ( N i ) . It is the content of the
Prym-Green Conjecture [FL] that K i, ( C, K C ⊗ η ) = 0 for ageneral Prym curve [ C, η ] ∈ R i +6 . This suggests that curves on Nikulin surfaces can berecognized by extra syzygies of their Prym-canonical embedding.Our initial motivation for considering Nikulin surfaces was to use them for thebirational classification of moduli spaces of even theta-characteristics and we propose toturn our attention to the moduli space S + g of even spin curves classifying pairs [ C, η ] ,where [ C ] ∈ M g is a smooth curve of genus g and η ∈ Pic g − ( C ) is an even theta-characteristic. Let S + g be the coarse moduli space associated to the Deligne-Mumfordstack of even stable spin curves of genus g , cf. [Cor]. The projection π : S + g → M g extends to a finite covering π : S + g → M g branched along the boundary divisor ∆ of M g . It is shown in [F] that S + g is a variety of general type as soon as g ≥ .The existence of the dominant morphism χ g : P N g → R g when g ≤ and g = 6 ,leads to a straightforward uniruled parametrization of S + g , which we briefly describe. Letus start with a general even spin curve [ C, η ] ∈ S + g and a non-trivial point of order two e C ∈ Pic ( C ) in the Jacobian, such that h ( C, e C ⊗ η ) ≥ . Since the curve [ C ] ∈ M g isgeneral, it follows that h ( C, e C ⊗ η ) = 1 and Z := supp( e C ⊗ η ) consists of g − distinctpoints. Applying Theorem 0.2, if g = 6 there exists a Nikulin K surface ( S, e ) containing C such that e C = e ⊗ O C . When g = 6 , there exists an Enriques surface ( S, e ) satisfyingthe same property, see [V1], and the construction described below goes through in thatcase as well. In the embedding φ |O S ( C ) | : S → P g , the span h Z i ⊂ P g is a codimension linear subspace and h ( S, I Z/S (1)) = 2 . Let P := P H (cid:0) S, I Z/S (1) (cid:1) ⊂ |O S ( C ) | be the corresponding pencil of curves on S . Each curve D ∈ P is endowed with the oddtheta-characteristic O D ( Z ) . Twisting this line bundle with e ⊗ O D ∈ Pic ( D ) , we obtainan even theta-characteristic on D . This procedure induces a rational curve in moduli m : P → S + g , P ∋ D [ D, e ⊗ O D ( Z )] , which passes through the general point [ C, η ] ∈ S + g . This proves the following result: Theorem 0.7.
The moduli space S + g is uniruled for g ≤ . It is known [F] that S + g is of general type when g ≥ . We complete the birationalclassification of S + g and wish to highlight the following result, see Theorem 0.1: ODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES 5 S +8 is a variety of Calabi-Yau type.We observe the curious fact that S − is unirational [FV] whereas S +8 is not evenuniruled. In contrast to the case of S ∓ g , the birational classification of other importantclasses of moduli spaces is not complete. The Kodaira dimension of M g is unknown for ≤ g ≤ , see [HM], [EH1], the birational type of R g is not understood in the range ≤ g ≤ , see [FL], whereas finding the Kodaira dimension of A is a notorious openproblem. Settling these outstanding cases is expected to require genuinely new ideas.The proof of Theorem 0.1 relies on two main ideas: Following [F], one finds an explicit effective representative for the canonical divisor K S +8 as a Q -combination of thedivisor Θ null ⊂ S +8 of vanishing theta-nulls, the pull-back π ∗ ( M , ) of the Brill-Noetherdivisor M , on M of curves with a g , and boundary divisor classes corresponding tospin curves whose underlying stable model is of compact type. This already implies theinequality κ ( S +8 ) ≥ . Each irreducible component of this particular representative of K S +8 is rigid (see Section 3), and the goal is to show that K S +8 is rigid as well. To that end,we use the existence of a birational model M of M inspired by Mukai’s work [M2]. Thespace M is realized as the following GIT quotient M := G (8 , ∧ V ) ss //SL ( V ) , where V = C . We note that ρ ( M ) = 1 and there exists a birational morphism f : M M , which contracts all the boundary divisors ∆ , . . . , ∆ as well as M , . Using the geomet-ric description of f , we establish a geometric characterization of points inside Θ null : Proposition 0.8.
Let C be a smooth curve of genus without a g . The following are equivalent: • There exists a vanishing theta-null L on C , that is, [ C, L ] ∈ Θ null . • There exists a smooth K surface S together with elliptic pencils | F | and | F | on S ,such that C ∈ | F + F | and L = O C ( F ) = O C ( F ) . The existence of such a doubly elliptic K surface S is equivalent to stating thatthere exists a smooth K extension S ⊂ P of the canonical curve C ⊂ P , such that therank three quadric C ⊂ Q ⊂ P which induces the theta-null L , lifts to a rank 4 quadric S ⊂ Q S ⊂ P . Having produced S , the pencils | F | and | F | define a product map φ : S → P × P , such that each smooth member D ∈ I := | φ ∗ O P × P (1 , | is a canonical curve containedin a rank quadric. A general pencil in I passing through C induces a rational curve R ⊂S +8 , and after intersection theoretic calculations on the stack S +8 , we prove the following: Proposition 0.9.
The theta-null divisor Θ null ⊂ S +8 is uniruled and swept by rational curves R ⊂ S +8 such that R · Θ null < and R · π ∗ ( M , ) = 0 . Furthermore R is disjoint from allboundary divisors π ∗ (∆ i ) for i = 1 , . . . , . G. FARKAS AND A. VERRA
Proposition 0.9 implies that K S +8 , expressed as a weighted sum of Θ null , the pull-back π ∗ ( M , ) and boundary divisors π ∗ (∆ i ) for i = 1 , . . . , , is rigid as well. Equiva-lently, κ ( S +8 ) = 0 . Note that since K S +8 consists of uniruled base components whichcan be blown-down, the variety S +8 is not minimal and there exists a birational model S of S +8 which is a genuine Calabi-Yau variety in the sense that K S = 0 . Finding an explicitmodular interpretation of this Calabi-Yau -fold (or perhaps even its equations!) is avery interesting question.1. P RYM - CANONICAL CURVES ON N IKULIN SURFACES
Let us start with a smooth K surface Y . A Nikulin involution on Y is an automor-phism ι ∈ Aut ( Y ) of order which is symplectic, that is, ι ∗ ( ω ) = ω , for all ω ∈ H , ( Y ) .A Nikulin involution has fixed points, see [Ni] Lemma 3, and the quotient ¯ Y := Y / h ι i has ordinary double point singularities. Let σ : ˜ S → Y be the blow-up of the fixedpoints and denote by E , . . . , E ⊂ ˜ S the exceptional divisors and by ˜ ι ∈ Aut ( ˜ S ) the au-tomorphism induced by ι . Then S := ˜ S/ h ˜ ι i is a smooth K surface and if f : ˜ S → S isthe projection, then N i := f ( E i ) are ( − -curves on S . The branch divisor of f is equal to N := P i =1 N i . We summarize the situation in the following diagram:(1) ˜ S σ −−−−→ Y f y y S −−−−→ ¯ Y Sometimes we shall refer to the pair ( Y, ι ) as a Nikulin surface, while keeping the previ-ous diagram in mind. We refer to [Mo], [vGS] for a lattice-theoretic study on the actionof the Nikulin involution on the cohomology H ( Y, Z ) = U ⊕ E ( − ⊕ E ( − , where U is the standard rank hyperbolic lattice and E is the unique even, negative-definiteunimodular lattice of rank . It follows from [Mo] Theorem 5.7 that the orthogonal com-plement E ( − ∼ = (cid:0) H ( Y, Z ) ι (cid:1) ⊥ is contained in Pic( Y ) , hence Y has Picard number atleast . The class O S ( N + · · · + N ) is divisible by , and we denote by e ∈ Pic( S ) theclass such that e ⊗ = O S ( N + · · · + N ) . Definition 1.1.
The
Nikulin lattice is an even lattice N of rank generated by elements { n i } i =1 and e := P i =1 n i , with the bilinear form induced by n i = − for i = 1 , . . . , and n i · n j = 0 for i = j .Note that N is the minimal primitive sublattice of H ( S, Z ) containing the classes N , . . . , N and e . For any Nikulin surface one has an embedding N ⊂ Pic( S ) . Assum-ing that ( Y, ι ) defines a general point in an irreducible component of the moduli space ofNikulin involutions, both Y and S have Picard number 9 and there is a decompositionPic ( S ) = Z · [ C ] ⊕ N , where C is an integral curve of genus g ≥ . According to [vGS]Proposition 2.2, only two cases are possible: either C · e = 0 so that the previous decom-position is an orthogonal sum, or else, C · e = 0 , this second case being possible onlywhen g is odd. In this paper we consider only Nikulin surfaces of the first kind.We fix an integer g ≥ and consider the lattice Λ g := Z · c ⊕ N , where c · c = 2 g − . Definition 1.2. A Nikulin surface of genus g is a K surface S together with a primitiveembedding of lattices j : Λ g ֒ → Pic( S ) such that C := j ( c ) is a nef class. ODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES 7
The coarse moduli space F N g of Nikulin surfaces of genus g is the quotient of the -dimensional domain D Λ g := { ω ∈ P (Λ g ⊗ Z C ) : ω = 0 , ω · ¯ ω > } by an arithmetic subgroup of O (Λ g ) . Its existence follows e.g. from [Do1] Section 3.We now consider a Nikulin surface f : ˜ S → S , together with a smooth curve C ⊂ S of genus g such that C · N = 0 . If ˜ C := f − ( C ) , then f C := f | ˜ C : ˜ C → C is an ´etaledouble covering. By the Hodge index theorem, ˜ C cannot split in two disjoint connectedcomponents, hence f C is non-trivial and e C := O C ( e ) ∈ Pic ( C ) is the non trivial 2-torsion element defining the covering f C . We set H ≡ C − e ∈ N S ( S ) , hence H = 2 g − and H · C = 2 g − . For further reference we collect a few easy facts: Lemma 1.3.
Let [ S, e, O S ( C )] ∈ F N g be a Nikulin surface such that Pic( S ) = Λ g . The followingstatements hold: (i) H i ( S, e ) = 0 for all i ≥ . (ii) Cliff( C ) = [ g − ] . (iii) The line bundle O S ( H ) is ample for g ≥ and very ample for g ≥ . In this range,it defines an embedding φ H : S → P g − such that the images φ H ( N i ) are lines for all i = 1 , . . . , . (iv) If g ≥ , the ideal of the surface Φ H ( S ) ⊂ P g − is cut out by quadrics.Proof. Recalling that e ⊗ = O S ( N + · · · + N ) and that the curves { N i } i =1 are pairwisedisjoint, it follows that H ( S, e ) = 0 and clearly H ( S, e ) = 0 . Since e = − , by Riemann-Roch one finds that H ( S, e ) = 0 as well.In order to prove (ii) we assume that
Cliff( C ) < [ g − ] . From [GL2] it follows thatthere exists a divisor D ∈ Pic( S ) such that h i ( S, O C ( D )) ≥ for i = 0 , and C · D ≤ g − ,such that O C ( D ) computes the Clifford index of C , that is, Cliff( C ) = Cliff( O C ( D )) . But C · ℓ ≡ g − for every class ℓ ∈ Pic( S ) , hence no such divisor D can exist.Moving to (iii), the ampleness (respectively very ampleness) of O S ( H ) is proved in[GS] Proposition 3.2 (respectively Lemma 3.1). From the exact sequence −→ O S ( − H ) −→ O S ( e ) −→ O C ( e ) −→ , one finds that h ( S, O S ( H )) = 0 and then dim | H | = g − . Furthermore H · N i = 1 for i = 1 , . . . , and the claim follows.To prove (iv), following [SD] Theorem 7.2, it suffices to show that there exists noirreducible curve Γ ⊂ S with Γ = 0 and H · Γ = 3 . Assume by contradiction that Γ ≡ aC − b N −· · ·− b N is such a curve, where necessarily a, b i ∈ Z ≤ . Then P i =1 b i = 2 ag − a − and P i =1 b i = a ( g − . Applying the Cauchy-Schwarz inequality (cid:0)P i =1 b i (cid:1) ≤ (cid:0)P i =1 b i (cid:1) , we obtain an immediate contradiction. (cid:3) We consider the P g -bundle p g : P N g → F N g , as well as the map χ g : P N g → R g , χ g ([ S, e, C ]) := [
C, e C := e ⊗ O C ] defined in the introduction. We fix a Nikulin surface [ S, e, O S ( C )] ∈ P N g . A Lefschetzpencil of curves { C λ } λ ∈ P inside |O S ( C ) | induces a rational curve Ξ g := { [ C λ , e C λ := e ⊗ O C λ ] : λ ∈ P } ⊂ R g . G. FARKAS AND A. VERRA
In the range where χ g is a dominant map, Ξ g is a rational curve passing through a generalpoint of R g , and it is of some interest to compute its numerical characters. If π : R g → M g denotes the projection map, we recall the formula [FL] Example 1.4(2) π ∗ ( δ ) = δ ′ + δ ′′ + 2 δ ram0 , where δ ′ := [∆ ′ ] , δ ′′ := [∆ ′′ ] and δ ram0 := [∆ ram0 ] are boundary divisor classes on R g whose meaning we recall. Let us fix a general point [ C xy ] ∈ ∆ induced by a -pointedcurve [ C, x, y ] ∈ M g − , and the normalization map ν : C → C xy , where ν ( x ) = ν ( y ) .A general point of ∆ ′ (respectively of ∆ ′′ ) corresponds to a stable Prym curve [ C xy , η ] ,where η ∈ Pic ( C xy )[2] and ν ∗ ( η ) ∈ Pic ( C ) is non-trivial (respectively, ν ∗ ( η ) = O C ). Ageneral point of ∆ ram0 is of the form [ X, η ] , where X := C ∪ { x,y } P is a quasi-stable curve,whereas η ∈ Pic ( X ) is characterized by η P = O P (1) and η ⊗ C = O C ( − x − y ) . Proposition 1.4. If Ξ g ⊂ R g is the curve induced by a pencil on a Nikulin surface, then Ξ g · λ = g + 1 , Ξ g · δ ′ = 6 g + 2 , Ξ g · δ ′′ = 0 and Ξ g · δ ram0 = 8 . It follows that Ξ g · K R g = g − .Proof. We use [FP] Lemma 2.4 to find that Ξ g · λ = π ∗ (Ξ g ) · λ = g + 1 and Ξ g · π ∗ ( δ ) = π ∗ (Ξ g ) · δ = 6 g + 18 , as well as Ξ g · π ∗ ( δ i ) = 0 for ≤ i ≤ [ g/ . For each ≤ i ≤ , thesublinear system P H ( O S ( C − N i )) ⊂ P H ( O S ( C )) intersects Ξ g transversally in onepoint which corresponds to a curve N i + C i ∈ |O S ( C ) | , where N i · C i = − N i = 2 and C i ≡ C − N i . Furthermore e ⊗ O N i = O N i (1) and e ⊗ C i = O C i ( − N i · C i ) . Each of thesepoints lie in the intersection Ξ g ∩ ∆ ram0 . All remaining curves in Ξ g are irreducible, hence Ξ g · δ ram0 = 8 . Since Ξ g · δ ′′ = 0 , from (2) we find that Ξ g · δ ′ = 6 g + 2 . Finally, accordingto [FL] Theorem 1.5 the formula K R g ≡ λ − δ ′ + δ ′′ ) − δ ram0 − · · · ∈ Pic( R g ) holds,therefore putting everything together, Ξ g · K R g = g − . (cid:3) The calculations in Proposition 1.4 are applied now to show that syzygies of Prym-canonical curves on Nikulin surfaces are exceptional when compared to those of generalPrym-canonical curves. To make this statement precise, let us recall the
Prym-Green Con-jecture , see [FL] Conjecture 0.7: If g := 2 i + 6 with i ≥ , then the locus U g,i := { [ C, η ] ∈ R i +6 : K i, ( C, K C ⊗ η ) = 0 } is a virtual divisor , that is, the degeneracy locus of two vector bundles of the same rank de-fined over R i +6 . The statement of the Prym-Green Conjecture is that this vector bundlemorphism is generically non-degenerate: Prym-Green Conjecture: K i, ( C, K C ⊗ η ) = 0 for a general Prym curve [ C, η ] ∈ R i +6 .The conjecture is known to hold in bounded genus and has been used in [FL] toshow that R g is of general type when g ≥ is even. Theorem 1.5.
For each [ S, e, C ] ∈ P N i +6 one has K i, ( C, K C ⊗ e C ) = 0 . In particular, thePrym-Green Conjecture fails along the locus N i +6 .Proof. If the non-vanishing K i, ( C, K C ⊗ η ) = 0 holds for a general point [ C, η ] ∈ R g , thenthere is nothing to prove, hence we may assume that U g,i is a genuine divisor on R g . The ODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES 9 class of its closure inside R g has been calculated [FL] Theorem 0.6: U g,i ≡ (cid:18) i + 2 i (cid:19) (cid:16) i + 7) i + 3 λ − δ ram0 − δ ′ − α δ ′′ − · · · (cid:17) ∈ Pic( R i +6 ) . From Proposition 1.4, by direct calculation one finds that Ξ g · U g,i = − (cid:0) i +3 i (cid:1) < , thus Ξ g ⊂ U g,i . By varying Ξ g inside R g , we obtain that N g ⊂ U g,i , which ends the proof. (cid:3) Remark 1.6.
A geometric proof of Theorem 1.5 using the Lefschetz hyperplane principlefor Koszul cohomology is given in [AF] Theorem 3.5. The indirect proof presented hereis however shorter and illustrates how cohomology calculations on R g can be used toderive geometric consequences for individual Prym curves. Remark 1.7.
One might ask whether similar applications to R g can be obtained usingEnriques surfaces. There is a major difference between Prym curves [ C, η ] ∈ R g lyingon a Nikulin surface and those lying on an Enriques surface. For instance, if C ⊂ S is acurve of genus g lying on an Enriques surface S , then from [CD] Corollary 2.7.1gon ( C ) ≤ (cid:8) F · C : F ∈ Pic( S ) , F = 0 , F (cid:9) ≤ p g − . In particular, for g sufficiently high, C is far from being Brill-Noether general. On theother hand, we have seen that for [ S, e, C ] ∈ P N g such that Pic( S ) = Λ g , one has thatgon ( C ) = [ g +32 ] . For this reason, the Prym-Nikulin locus N g := Im ( χ g ) ⊂ R g appears asa more promising and less constrained locus than the Prym-Enriques locus in R g , beingtransversal to stratifications of R g coming from Brill-Noether theory.2. T HE P RYM -N IKULIN LOCUS IN R g FOR g ≤ In this section we give constructive proofs of Theorems 0.2 and 0.4. Comparingthe dimensions dim ( P N g ) = 11 + g and dim ( R g ) = 3 g − , one may inquire whether themorphism χ g : P N g → R g is dominant when g ≤ . The similar question for ordinary K surfaces has been answered by Mukai [M1]. Let F g denote the -dimensional modulispace of polarized K surfaces of genus g and consider the associated P g -bundle P g := (cid:8) [ S, C ] : C ⊂ S is a smooth curve such that [ S, O S ( C )] ∈ F g (cid:9) . The map q g : P g → M g forgetting the K surface is dominant if and only if g ≤ and g = 10 . The result for g = 10 is contrary to untutored expectation since the general fibreof q is -dimensional, hence dim ( Im ( q )) = dim ( P ) − . A strikingly similarpicture emerges for Nikulin surfaces and Prym curves. The morphism χ g : P N g → R g isdominant when g ≤ and g = 6 . For each genus we describe a geometric constructionthat furnishes a Nikulin surface in the fibre χ − g (cid:0) [ C, η ] (cid:1) over a general point [ C, η ] ∈ R g .2.1. Nikulin surfaces of genus 7.
We start with a general element [ C, η ] ∈ R and con-struct a Nikulin surface containing C . One may assume that gon ( C ) = 5 and that theline bundle η does not lie in the difference variety C − C ⊂ Pic ( C ) , or equivalently,the linear series L := K C ⊗ η ∈ W ( C ) is very ample. It is a consequence of [GL1]Theorem 2.1 that the Prym-canonical image C | L | −→ P is quadratically normal, that is, h ( P , I C/ P (2)) = 3 . Lemma 2.1.
For a general [ C, η ] ∈ R , the base locus of |I C/ P (2) | is a smooth K3 surface. Proof.
The property that the base locus of |I C/ P (2) | is smooth, is open in R and it sufficesto exhibit a single Prym-canonical curve [ C, η ] ∈ R satisfying it. Let us fix an element ( S, e, C ) ∈ P N such that Pic( S ) = Λ and set H ≡ C − e . Then according to Lemma1.3, φ H : S → P is an embedding whose image φ H ( S ) is ideal-theoretically cut out byquadrics. Moreover gon ( C ) = 5 , hence K C ⊗ e C ∈ W ( C ) is quadratically normal. Thisimplies that H ( S, O S (2 H − C )) = H ( S, O S (2 H − C )) = 0 , and then H ( P , I S/ P (2)) ∼ = H ( P , I C/ P (2)) , therefore the quadrics in |I C/ P (2) | cut out precisely the surface S . (cid:3) Remark 2.2.
This proof shows that if [ S, e, C ] ∈ P N is general then χ − (cid:0) [ C, e C ] (cid:1) = [ S, e, C ] and in particular the fibre χ − ( (cid:2) C, e C ] (cid:1) is reduced. Indeed, let [ S ′ , e ′ , C ] ∈ P N be anarbitrary Nikulin surface containing C . Set H ′ ≡ C − e ′ ∈ N S ( S ′ ) . We may assume that Pic( S ′ ) = Λ , therefore the map φ H ′ : S ′ → P is an embedding whose image is cut outby quadrics. Since Cliff ( C ) = 3 , from Lemma 1.3 we find that K C ⊗ e C is quadraticallynormal and then S ′ is cut out by the quadrics contained in Prym-canonical embeddingof C ⊂ P .Since both P N and R are irreducible varieties of dimension , Remark 2.2 showsthat χ : P N → R is a birational morphism and we now describe χ − . Proposition 2.3.
For a general [ C, η ] ∈ R , the surface S := bs |I C/ P (2) | is a polarized Nikulinsurface of genus .Proof. We show that
Pic( S ) ⊃ Z · C ⊕ N . Denote by H ⊂ S the hyperplane class and let N : ≡ C − H ) , thus N = − , N · H = 8 and N · C = 0 . We aim to prove that N islinearly equivalent to a sum of pairwise disjoint integral ( − curves on S . We considerthe following exact sequence −→ O S ( N − C ) −→ O S ( N ) −→ O C ( N ) −→ . Note that O C ( N ) is trivial because e C = O C ( C − H ) and that h ( S, O S ( N − C )) = h ( S, O S ( C − H )) = 0 , because C ⊂ P is quadratically normal. Passing to the longexact sequence, it follows that h ( S, O S ( N )) = 1 . Using Remark 2.2 it follows that N ≡ N + · · · + N , where N i · N j = − δ ij . Finally, to conclude that [ S, Z · C ⊕ N ] ∈ F N we must show that there is a primitive embedding Z · C ⊕ N ֒ → Pic ( S ) . We apply [vGS]Proposition 2.7. Since H ( ˜ S, O S ( ˜ C )) = H ( S, O S ( C )) ⊕ H ( S, O S ( C ) ⊗ e ∨ ) and sectionsin the second summand vanish on the exceptional divisor of the morphism σ : ˜ S → Y ,it follows that this is precisely the decomposition of H ( Y, O Y ( ˜ C )) into ι ∗ Y -eigenspaces.Invoking loc. cit. , we finish the proof. (cid:3) The symmetric determinantal cubic hypersurface and Prym curves.
We provide ageneral set-up that allows us to reconstruct a Nikulin surface from a Prym curve of genus g ≤ . Let us start with a curve [ C, η ] ∈ R g inducing an ´etale double cover f : ˜ C → C together with an involution ι : ˜ C → ˜ C such that f ◦ ι = f . For each integer r ≥ − , the Prym-Brill-Noether locus is defined as the locus V r ( C, η ) := { L ∈ Pic g − ( ˜ C ) : Nm f ( L ) = K C , h ( L ) ≥ r + 1 and h ( L ) ≡ r + 1 mod 2 } . Note that V − ( C, η ) = Pr(
C, η ) . For each line bundle L ∈ V r ( C, η ) , the Petri map µ ( L ) : H ( ˜ C, L ) ⊗ H ( ˜ C, K ˜ C ⊗ L ∨ ) → H ( ˜ C, K ˜ C ) ODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES 11 splits into an ι -anti-invariant part µ − ( L ) : Λ H ( ˜ C, L ) → H ( C, K C ⊗ η ) , s ∧ t s · ι ∗ ( t ) − t · ι ∗ ( s ) , and an ι -invariant part respectively µ +0 ( L ) : Sym H ( ˜ C, L ) → H ( C, K C ) , s ⊗ t + t ⊗ s s · ι ∗ ( t ) + t · ι ∗ ( s ) . For a general [ C, η ] ∈ R g , the Prym-Petri map µ − ( L ) is injective for every L ∈ V r ( C, η ) and V r ( C, η ) is equidimensional of dimension g − − (cid:0) r +12 (cid:1) , see [We]. We introduce the universal Prym-Brill-Noether variety R rg := n(cid:0) [ C, η ] , L (cid:1) : [ C, η ] ∈ R g , L ∈ V r ( C, η ) o . When g − − (cid:0) r +12 (cid:1) ≥ , the variety R rg is irreducible of dimension g − − (cid:0) r +12 (cid:1) . Wepropose to focus on the case r = 2 and g ≥ and choose a general triple ( f : ˜ C → C, L ) ∈R g , such that L is base point free and h ( ˜ C, L ) = 3 .Setting P := P (cid:0) H ( L ) ∨ (cid:1) , we consider the quasi-´etale double cover q : P × P → P obtained by projecting via the Segre embedding to the space of symmetric tensors. Notethat q is ramified along the diagonal ∆ ⊂ P × P and V := q (∆) ⊂ P is the Veronesesurface. Moreover Σ := Im ( q ) is the determinantal symmetric cubic hypersurface iso-morphic to the secant variety of V . We have the following commutative diagram: ˜ C ( L,ι ∗ L ) / / f (cid:15) (cid:15) P × P q (cid:15) (cid:15) , , XXXXXXXXXXXXXXXX P = P (cid:0) H ( L ) ∨ ⊗ H ( L ) ∨ (cid:1) r r e e e e e e e C µ +0 ( L ) / / P = P (Sym H ( L ) ∨ ) Observe that the involution ι : P → P given by ι [ v ⊗ w ] := [ w ⊗ v ] where v, w ∈ H ( L ) , iscompatible with ι : ˜ C → ˜ C . To summarize, giving a point ( ˜ C → C, L ) ∈ R g is equivalentto specifying a symmetric determinantal cubic hypersurface Σ ∈ H ( P g − , I C/ P g − (3)) containing the canonical curve.2.3. A birational model of F N . As a warm-up, we indicate how the set-up describedabove is a generalization of the construction that Catanese [Ca] used to prove that R isrational. For a general point [ C, η ] ∈ R we find that V ( C, η ) = { L, ι ∗ L } , that is, the pair ( L, ι ∗ L ) is uniquely determined. The map µ ( L ) has corank and P C := P (cid:0) Im µ ( L ) (cid:1) ⊂ P has codimension . The intersection ˜ T := ( P × P ) ∩ P C is a del Pezzo surface of degree , whereas T := Σ ∩ P is a -nodal Cayley cubic. Here we set P := P (cid:0) H ( K C ) ∨ (cid:1) . Thedouble cover q : ˜ T → T is ramified at the singular points of T .To obtain a Nikulin surface containing [ C, η ] , we reverse this construction and startwith a quartic rational normal curve R ⊂ P and denote by ¯ Y := Sec( R ) ⊂ P its secantvariety, which we view as a hyperplane section of Σ ⊂ P . Retaining the notation ofdiagram (1), for a general quadric Q ∈ |O P (2) | , the intersection ¯ Y := ¯ Y ∩ Q is a K surface with rational double points at R ∩ Q . There exists a cover q : Y → ¯ Y ramifiedat the singular points of Y , induced by restriction from the map q : P × P → Σ . Clearly q : Y → ¯ Y is a Nikulin covering, and a hyperplane section in |O ¯ Y (1) | induces a Prymcurve [ C, η ] ∈ R having general moduli. Moreover we have a birational isomorphism F N ∼ = P (cid:16) H ( O P (2)) (cid:17) ss //SL , where P GL = Aut ( R ) ⊂ P GL . An immediate consequence is that F N is unirational.2.4. Nikulin surfaces of genus . We prove that χ : P N → R is dominant and fix acomplete intersection of quadrics Y ⊂ P invariant with respect to an involution fixinga line L ⊂ P and a -dimensional linear subspace Λ ⊂ P . The projection π L : P Λ induces a quartic ¯ Y := π L ( Y ) with nodes, which is a Nikulin surface. We check thata general Prym curve [ C, η ] ∈ R corresponding to an ´etale cover f : ˜ C → C embeds insuch a surface.Indeed, the canonical model ˜ C ⊂ P is a complete intersection of quadrics. Fixingprojective coordinates on P , we can assume that the involution ι : ˜ C → ˜ C is inducedby the projective involution [ x : y : u : v : t ] ↔ [ − x : − y : u : v : t ] . Note that the ι ∗ anti-invariant quadratic forms are vectors q = ax + by , where a, b are linear forms in u, v, t . Since ˜ C is complete intersection of 3 quadrics, no non-zero quadric q = ax + by vanishes on ˜ C , for not, ˜ C would intersect the plane { x = y = 0 } and then ι would havefixed points. Thus ι acts as the identity on the space H ( P , I ˜ C/ P (2)) . Hence it follows ˜ C = { a + b = a + b = a + b = 0 } , where a i , b i are quadratic forms in x, y and u, v, t .Passing to P by adding one coordinate h , we can choose quadratic forms a i + b i + hl i ,where l i is a general linear form in h, u, v, t . Consider the surface Y ⊂ P defined by thelatter 3 equations. Then [ x : y : h : u : v : t ] ↔ [ − x : − y : h : u : v : t ] induces a Nikulininvolution on Y . Let π L : Y → P be the projection of center L = { h = u = v = t = 0 } .Then Y := π L ( Y ) is a quartic Nikulin surface and C = π L ( ˜ C ) is a plane section of it.2.5. Nikulin surfaces of genus . To describe the morphism χ : P N → R more ge-ometrically, we use the set-up introduced in Subsection 2.2. If [ C, η ] ∈ R is general,then dim V ( C, η ) = 1 , the ι -invariant Petri map µ − ( L ) is injective, µ +0 ( L ) surjective, thusdim (cid:0) Coker µ ( L ) (cid:1) = 1 . We consider the hyperplane P C := P (cid:0) Im ( µ ( L ) (cid:1) ⊂ P (cid:0) H ( L ) ∨ ⊗ H ( L ) ∨ (cid:1) and also set P := P (cid:0) H ( K C ) ∨ (cid:1) ⊂ P . Then we further denote ˜ T := ( P × P ) ∩ P C and T := Σ ∩ P . Note that ˜ T is a degree threefold in P C . Since the hyperplane P C is ι -invariant, it follows ˜ T is also endowed with the involution ι ˜ T ∈ Aut ( ˜ T ) such that Fix ( ι ˜ T ) = ∆ ∩ ˜ T is a rationalquartic curve in P . Furthermore T ⊂ P is the secant variety of R . Proposition 2.4.
For a general point [ C, η, L ] ∈ R the following statements hold: (i) The threefold ˜ T ⊂ P × P is smooth, while T ⊂ P is singular precisely along R . (ii) h ( ˜ T , I ˜ C/ ˜ T (2)) = 3 . Moreover H i ( ˜ T , I ˜ C/ ˜ T (2)) = 0 for i = 1 , . (iii) Every quadratic section in the linear system |I ˜ C/ ˜ T (2) | is ι -invariant, that is, H ( ˜ T , I ˜ C/ ˜ T (2)) = q ∗ H ( T, I C/T (2)) . ODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES 13 (iv)
A general quadratic section Y ∈ |I ˜ C/ ˜ T (2) | is a smooth K surface endowed with aninvolution ι Y with fixed points precisely at the points in the intersection R ∩ Y .Proof. We take cohomology in the following exact sequence −→ I ˜ C/ P × P (2) −→ O P × P (2) −→ K ⊗ C −→ , to note that h ( I ˜ C/ ˜ T (2)) = 3( ⇔ H ( I ˜ C/ P × P (2)) = 0 ), if and only if the composed map Sym H ( ˜ C, L ) ⊗ Sym H ( ˜ C, ι ∗ L ) → H ( ˜ C, L ⊗ ) ⊗ H ( ˜ C, ι ∗ ( L ⊗ )) → H ( ˜ C, K ⊗ C ) is surjective. This is an open condition and a triple ( ˜ C f → C, L ) ∈ R g satisfying it, and forwhich moreover ˜ T ⊂ P × P is smooth, has been constructed in [V2] Section 4. Finally,from the exact sequence −→ I T/ P (2) −→ I C/ P (2) −→ I C/T (2) → , we compute that h ( T, I C,T (2)) = 3 , therefore q ∗ : H ( T, I C/T (2)) → H ( ˜ T , I ˜ C/ ˜ T (2)) isan isomorphism, based on dimension count. Part (iv) is a consequence of (i)-(iii). Assumethat ¯ Y = T ∩ Q , where Q ∈ H ( I C/ P (2)) . Then Y = ˜ T ∩ q ∗ ( Q ) and ¯ Y is the quotientof Y by the involution ι Y obtained by restriction from ι ∈ Aut ( P × P ) . It follows thatthe covering q : Y → ¯ Y is a Nikulin surface such that C ⊂ ¯ Y ⊂ P . To conclude, wemust check that for a general choice of Y ∈ |I ˜ C/ ˜ T (2) | , the point [ Y, ι Y ] gives rise to anelement of F N , that is, using the notation of diagram (1), that Pic ( S ) = Λ . Proposition2.7 from [vGS] picks out two possibilities for Pic ( Y ) (or equivalently for Pic ( S ) ), and wemust check that Z · O Y ( ˜ C ) ⊕ E ( − has index in Pic ( Y ) , see also [GS] Corollary 2.2. This is achieved by finding the decomposition of H ( O Y ( ˜ C )) into ι ∗ Y -eigenspaces.In the course of the proof of [V2] Proposition 5.2 an example of a smooth quadratic section Y ∈ |I ˜ C/ ˜ T (2) | is constructed such that H ( Y, O Y ( ˜ C )) + = q ∗ H ( ¯ Y , O ¯ Y ( C )) . In particular the (+1) -eigenspace of H ( Y, O Y ( ˜ C )) is -dimensional and invoking oncemore [vGS] Proposition 2.7, we conclude that [ Y, ι Y ] ∈ F N . (cid:3) We close this subsection with an amusing result on a geometric divisor on R . Fora Prym curve [ C, η ] ∈ R and L := K C ⊗ η ∈ W ( C ) , we observe that the vector spacesentering the multiplication map ν ( L ) : Sym H ( C, L ) → H ( C, L ⊗ ) have the samedimension. The condition that ν ( L ) be not an isomorphism is divisorial in R . We havenot been able to find a direct proof of the following equality of cycles on R , even thoughone inclusion is straightforward: Theorem 2.5.
Let [ C, η ] ∈ R be a Prym curve such that the Prym-canonical line bundle K C ⊗ η is very ample. Then φ K C ⊗ η : C → P lies on a cubic surface if and only if C is trigonal.Proof. Let D be the locus of Prym curves whose Prym-canonical model lies on a cubic D := { [ C, η ] ∈ R : ν ( ω C ⊗ η ) : Sym H ( C, ω C ⊗ η ) ≇ −→ H ( C, ω ⊗ C ⊗ η ⊗ ) } , We are grateful to the referee for raising this point that we have initially overlooked. and D the closure inside R of the divisor { [ C, η ] ∈ R : η ∈ C − C } of smooth Prymcurves for which L := K C ⊗ η ∈ W ( C ) is not very ample. Obviously, D − D ≥ , forif L is not very ample, then the multiplication map ν ( L ) : Sym H ( C, L ) → H ( C, L ⊗ ) cannot be an isomorphism. The class of D can be read off [FL] Theorem 5.2: D ≡ λ − δ ′ + δ ′′ ) − δ ram0 − · · · ∈ Pic( R ) . For i ≥ , let E i be the vector bundle over R with fibre E i [ C, η ] = H ( C, ω ⊗ iC ⊗ η ⊗ i ) forevery [ C, η ] ∈ R g . One has the following formulas from [FL] Proposition 1.7: c ( E i ) = (cid:18) i (cid:19) (12 λ − δ ′ − δ ′′ − δ ram0 ) + λ − i δ ram0 ∈ Pic( R g ) . As a consequence, D ≡ c ( E ) − c (Sym E ) ≡ λ − δ + δ ′′ ) − δ ram0 − · · · ∈ Pic( R ) , therefore D − D ≡ λ − ( δ ′ + δ ′′ ) − δ ram0 − · · · = π ∗ (8 λ − δ − · · · ) ≥ , where theterms left out are combinations of boundary divisors π ∗ ( δ i ) with i ≥ , corresponding toreducible curves. The only effective divisors D ≡ aλ − b δ − b δ − b δ on M such that ab ≤ and satisfying ∆ i * supp( D ) for i = 1 , , are multiples of the trigonal locus M , (the proof is identical to that of Proposition 5.1). This proves that if [ C, η ] ∈ D − D , with C being a smooth curve, then necessarily [ C ] ∈ M , , which finishes the proof. (cid:3)
3. A
SINGULAR QUADRATIC COMPLEX AND A BIRATIONAL MODEL FOR F N Let us set V := C n +1 and denote by G := G (2 , V ) ⊂ P ( ∧ V ) the Grassmannian oflines in P ( V ) . We fix once and for all a smooth quadric Q ⊂ P ( V ) . The projective tangentbundle P Q := P ( T Q ) can be realized as the incidence correspondence P Q = (cid:8) ( x, ℓ ) ∈ Q × G : x ∈ ℓ ⊂ P ( T x Q ) (cid:9) . For each point x ∈ Q , the fibre P Q ( x ) is the space of lines tangent to Q at x . We introducethe projections p : P Q → G and q : P Q → Q , then set W Q := p ( P Q ) = { ℓ ∈ G : ℓ is tangent to the quadric Q } . Note that W Q contains the Hilbert scheme of lines in Q , which we denote by V Q ⊂ W Q .It is well-known that V Q is smooth, irreducible and dim ( V Q ) = 2 n − . The restriction p | p − ( W Q − V Q ) is an isomorphism and E Q := p − ( V Q ) ⊂ P Q is the exceptional divisor of p . Proposition 3.1.
The variety W Q is a quadratic complex of lines in G . Its singular locus is equalto V Q and each point of V Q is an ordinary double point of W Q .Proof. Let Q : V → C be the quadratic form whose zero locus is the quadric hypersurfacealso denoted by Q ⊂ P ( V ) , and ˜ Q : V × V → C the associated bilinear map. We definethe bilinear map ν ( ˜ Q ) : ∧ V × ∧ V → C by the formula ν ( ˜ Q )( u ∧ v, s ∧ t ) := ˜ Q ( u, s ) ˜ Q ( v, t ) − ˜ Q ( v, s ) ˜ Q ( u, t ) for u, v, s, t ∈ V , and denote by ν ( Q ) : ∧ V → C the induced quadratic form.For fixed points x = [ u ] ∈ Q and y = [ v ] ∈ P ( V ) , we observe that the line ℓ = h x, y i is tangent to Q if and only if ˜ Q ( u, v ) = 0 ⇔ ν ( Q )( u ∧ v ) = 0 . Therefore W Q = G ∩ ν ( Q ) is a quadratic line complex in G , being the vanishing locus of ν ( Q ) .Keeping the same notation, a point ℓ = [ u ∧ v ] ∈ W Q is a singular point, if and onlyif the linear form ν ( ˜ Q )( u ∧ v, − ) vanishes along P ( T ℓ G ) . Since P ( T ℓ G ) is spanned by ODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES 15 the Schubert cycle { m ∈ G : m ∩ ℓ = ∅} , any tangent vector in T ℓ ( G ) has a representativeof the form u ∧ a − v ∧ b , where a, b ∈ V . We obtain that [ u ∧ v ] ∈ Sing( W Q ) if and only if Q ( v, v ) = 0 , that is, ℓ = [ u ∧ v ] ∈ V Q . Since W Q is a quadratic complex, each point ℓ ∈ V Q has multiplicity . (cid:3) The map p : P Q → W Q appears as a desingularization of the quadratic complex W Q . We shall compute the class of the exceptional divisor E Q of P Q . Let H := p ∗ ( O G (1)) be the class of the family of tangent lines to Q intersecting a fixed ( n − -plane in P ( V ) and B := q ∗ ( O Q (1)) ∈ Pic( P Q ) . Furthermore, we consider the class h ∈ N S ( P Q ) of thepencil of tangent lines to Q with center a given point x ∈ Q . It is clear that h · H = 1 and h · B = 0 . If ℓ ∈ V Q is a fixed line, let s ∈ N S ( P Q ) be the class of the family { ( x, ℓ ) : x ∈ ℓ } . Then s · H = 0 and s · B = 1 . Lemma 3.2.
The linear equivalence E Q ≡ H − B in Pic( P Q ) holds. In particular, the class E Q is divisible by 2 and it is the branch divisor of a double cover f : ˜ P Q → P Q . Proof.
To compute the class of E Q it suffices to compute h · E Q and s · E Q . First we notethat h · E Q = 2 . Indeed a pencil of tangent lines to Q through a fixed point x ∈ Q hastwo elements which are in Q . Finally, recalling that V Q = Sing( W Q ) consists of ordinarydouble points, we obtain that s · E Q = − , since p − ( ℓ ) is a conic inside P (cid:0) N V Q / G ( ℓ ) (cid:1) . (cid:3) A birational model for F N . Let us now specialize to the case n = 4 , that is, Q ⊂ P , G = G (2 , ⊂ P and dim ( W Q ) = 5 . The class of V Q equals σ , ∈ H ( G , Z ) see [HP] p. 366, therefore deg ( V Q ) = 4 σ , · σ = 8 .This can also be seen by recalling that V Q is isomorphic to the Veronese -fold ν ( P ) ⊂ P .The double covering f : ˜ P Q → P Q constructed above has a transparent projectiveinterpretation. For ( x, ℓ ) ∈ P Q , we denote by Π ℓ ∈ G (3 , V ) the polar space of ℓ defined asthe base locus of the pencil of polar hyperplanes { z ∈ P ( V ) : ˜ Q ( y, z ) = 0 } y ∈ ℓ . Clearly x ∈ Π ℓ ⊂ P ( T x Q ) and Q ∩ Π ℓ is a conic of rank at most in Π ℓ . When ℓ ∈ W Q − V Q , thequadric has rank exactly which corresponds to a pair of lines ℓ + ℓ with ℓ , ℓ ∈ V Q . Thedouble cover is induced by the map from the parameter space of the lines themselves.In the next statement we shall keep in mind the notation of diagram (1): Proposition 3.3.
A general codimension linear section ¯ Y := Λ ∩ W Q of the quadratic complex W Q where Λ ∈ G (7 , ∧ V ) , is a -nodal K surface with desingularization p : S := p − ( ¯ Y ) → ¯ Y .
The triple [ S, O S ( H − B ) , O S ( H )] ∈ F N is a Nikulin surface of genus and the induced doublecover is the restriction f : ˜ S := f − ( S ) → S .Proof. We fix a general -plane Λ ∈ G (7 , ∧ V ) . Since K W Q = O W Q ( − H ) , by adjunctionwe obtain that ¯ Y := Λ ∩ W Q is a K surface. From Bertini’s theorem, ¯ Y has ordinarydouble points at the points of intersection Λ ∩ V Q . General hyperplane sections of C ∈ |O ¯ Y ( H ) | , viewed as codimension linear sections of W Q , are canonical curves of genus , endowed with a line bundle of order given by O C ( H − B ) . The remainingstatements are immediate. (cid:3) It turns out that the general Nikulin surface of genus arises in this way: Theorem 3.4.
Let V := C and Q ⊂ P ( V ) be a smooth quadric. One has a dominant map ϕ : G (7 , ∧ V ) ss // Aut( Q ) F N , given by ϕ (Λ) := (cid:2) S := p − (Λ ∩ W Q ) , O S ( H − B ) , O S ( H ) (cid:3) .Proof. Via the embedding Aut ( Q ) ⊂ P GL ( V ) ֒ → P GL ( ∧ V ) , we observe that everyautomorphism of Q induces an automorphism of P ( ∧ V ) that fixes both W Q and V Q .Since (i) the moduli space F N is irreducible and (ii) polarized Nikulin surfaces have finiteautomorphism groups, it suffices to observe that dim G (7 , ∧ V ) // Aut ( Q ) = 21 −
10 = 11 and dim ( F N ) = 11 as well. (cid:3) Corollary 3.5.
The Prym-Nikulin locus N ⊂ R is an irreducible unirational divisor, which isset-theoretically equal to the ramification locus of the Prym map Pr : R → A U , = { [ C, η ] ∈ R : K , ( C, K C ⊗ η ) = 0 } . Furthermore, the exists a dominant rational map G (6 , ∧ V ) ss // Aut( Q ) N . Proof.
Just observe that h C i = P and that this has codimension 4 in P ( ∧ V ) , hence thereis a P of Nikulin sections of W Q containing C . (cid:3) The divisor K ⊂ M of sections of K surfaces is known to be an extremal pointof the effective cone Eff ( M ) . An analogous result holds for the closure of N : Proposition 3.6.
The Prym-Nikulin divisor N is extremal in the effective cone Eff( R ) :Proof. It follows from [FL] Theorem 0.6 that N ≡ λ − δ ram0 − ( δ ′ + δ ′′ ) − · · · ∈ Pic( R ) .The divisor KN is filled-up by the rational curves Ξ ⊂ R constructed in the course ofproving Theorem 1.4. We compute that Ξ · N = − , which completes the proof. (cid:3)
4. S
PIN CURVES AND THE DIVISOR Θ null We turn our attention to the moduli space of spin curves and begin by setting no-tation and terminology. If M is a Deligne-Mumford stack, we denote by M its associatedcoarse moduli space. A Q -Weil divisor D on a normal Q -factorial projective variety X issaid to be movable if codim (cid:0)T m Bs | mD | , X (cid:1) ≥ , where the intersection is taken over all m which are sufficiently large and divisible. We say that D is rigid if | mD | = { mD } , forall m ≥ such that mD is an integral Cartier divisor. The Kodaira-Iitaka dimension of adivisor D on X is denoted by κ ( X, D ) .If D = m D + · · · + m s D s is an effective Q -divisor on X , with irreducible compo-nents D i ⊂ X and m i > for i = 1 , . . . , s , a (trivial) way of showing that κ ( X, D ) = 0 isby exhibiting for each ≤ i ≤ s , an irreducible curve Γ i ⊂ X passing through a generalpoint of D i , such that Γ i · D i < and Γ i · D j = 0 for i = j .We recall basic facts about the moduli space S + g and refer to [Cor], [F] for details. ODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES 17
Definition 4.1. An even spin curve of genus g consists of a triple ( X, η, β ) , where X is agenus g quasi-stable curve, η ∈ Pic g − ( X ) is a line bundle of degree g − such that η E = O E (1) for every rational component E ⊂ X with | E ∩ ( X − E ) | = 2 (such a componentis called exceptional ), h ( X, η ) ≡ mod , and β : η ⊗ → ω X is a morphism of sheaveswhich is generically non-zero along each non-exceptional component of X .Even spin curves of genus g form a smooth Deligne-Mumford stack π : S + g → M g .At the level of coarse moduli schemes, the morphism π : S + g → M g is the stabilizationmap π ([ X, η, β ]) := [st( X )] , which associates to a quasi-stable curve its stable model.We explain the boundary structure of S + g : If [ X, η, β ] ∈ π − ([ C ∪ y D ]) , where [ C, y ] ∈M i, , [ D, y ] ∈ M g − i, and ≤ i ≤ [ g/ , then necessarily X = C ∪ y E ∪ y D , where E is an exceptional component such that C ∩ E = { y } and D ∩ E = { y } . Moreover η = (cid:0) η C , η D , η E = O E (1) (cid:1) ∈ Pic g − ( X ) , where η ⊗ C = K C , η ⊗ D = K D . The condition h ( X, η ) ≡ mod , implies that the theta-characteristics η C and η D have the same parity.We denote by A i ⊂ S + g the closure of the locus corresponding to pairs ([ C, y, η C ] , [ D, y, η D ]) ∈ S + i, × S + g − i, and by B i ⊂ S + g the closure of the locus corresponding to pairs ([ C, y, η C ] , [ D, y, η D ]) ∈ S − i, × S − g − i, . We set α i := [ A i ] ∈ Pic( S + g ) , β i := [ B i ] ∈ Pic( S + g ) , and then one has the relation(3) π ∗ ( δ i ) = α i + β i . We recall the description of the ramification divisor of the covering π : S + g → M g .For a point [ X, η, β ] ∈ S + g corresponding to a stable model st ( X ) = C yq := C/y ∼ q ,with [ C, y, q ] ∈ M g − , , there are two possibilities depending on whether X possesses anexceptional component or not. If X = C yq (i.e. X has no exceptional component) and η C := ν ∗ ( η ) where ν : C → X denotes the normalization map, then η ⊗ C = K C ( y + q ) . Foreach choice of η C ∈ Pic g − ( C ) as above, there is precisely one choice of gluing the fibres η C ( y ) and η C ( q ) such that h ( X, η ) ≡ mod . We denote by A the closure in S + g of thelocus of spin curves [ C yq , η C ∈ p K C ( y + q )] as above.If X = C ∪ { y,q } E , where E is an exceptional component, then η C := η ⊗ O C is atheta-characteristic on C . Since H ( X, ω ) ∼ = H ( C, ω C ) , it follows that [ C, η C ] ∈ S + g − . Wedenote by B ⊂ S + g the closure of the locus of spin curves (cid:2) C ∪ { y,q } E, E ∼ = P , η C ∈ p K C , η E = O E (1) (cid:3) ∈ S + g . If α := [ A ] , β := [ B ] ∈ Pic ( S + g ) , we have the relation, see [Cor]:(4) π ∗ ( δ ) = α + 2 β . In particular, B is the ramification divisor of π . An important effective divisor on S + g isthe locus of vanishing theta-nulls Θ null := { [ C, η ] ∈ S + g : H ( C, η ) = 0 } . The class of its compactification inside S + g is given by the formula, cf. [F]:(5) Θ null ≡ λ − α − [ g/ X i =1 β i ∈ Pic( S + g ) . It is also useful to recall the formula for the canonical class of S + g : K S + g ≡ π ∗ ( K M g ) + β ≡ λ − α − β − [ g/ X i =1 ( α i + β i ) − ( α + β ) . An argument involving spin curves on certain singular canonical surfaces in P ,implies that for g ≤ , the divisor Θ null is uniruled and a rigid point in the cone of effectivedivisors Eff( S + g ) : Theorem 4.2.
For g ≤ the divisor Θ null ⊂ S + g is uniruled and rigid. Precisely, through ageneral point of Θ null there passes a rational curve Γ ⊂ S + g such that Γ · Θ null < . In particular,if D is an effective divisor on S + g with D ≡ n Θ null for some n ≥ , then D = n Θ null .Proof. A general point [ C, η C ] ∈ Θ null corresponds to a canonical curve C | K C | ֒ → P g − lyingon a rank quadric Q ⊂ P g − such that C ∩ Sing( Q ) = ∅ . The pencil η C is recoveredfrom the ruling of Q . We construct the pencil Γ ⊂ S + g by representing C as a section of anodal canonical surface S ⊂ Q and noting that dim |O S ( C ) | = 1 . The construction of S depends on the genus and we describe the various cases separately. (i) ≤ g ≤ . We choose V ∈ G (cid:0) , H ( C, K C ) (cid:1) such that if π V : P g − P ( V ∨ ) denotesthe projection, then ˜ Q := π V ( Q ) is a quadric of rank . Let C ′ := π V ( C ) ⊂ P ( V ∨ ) be theprojection of the canonical curve C . By counting dimensions we find thatdim n I C ′ / P ( V ∨ ) (2) := Ker (cid:8)
Sym ( V ) → H ( C, K ⊗ C ) (cid:9)o ≥ − g ≥ , that is, the embedded curve C ′ ⊂ P lies on at least independent quadrics, namely therank quadric ˜ Q and Q , Q , Q ∈ | I C ′ / P ( V ∨ ) (2) | . By choosing V sufficiently general wemake sure that S := ˜ Q ∩ Q ∩ Q ∩ Q is a canonical surface in P ( V ∨ ) with nodes cor-responding to the intersection T i =1 Q i ∩ Sing( ˜ Q ) (This transversality statement can alsobe checked with Macaulay by representing C as a section of the corresponding Mukaivariety). From the exact sequence on S , −→ O S −→ O S ( C ) −→ O C ( C ) −→ , coupled with the adjunction formula O C ( C ) = K C ⊗ K ∨ S | C = O C , as well as the fact H ( S, O S ) = 0 , it follows that dim | C | = 1 , that is, C ⊂ S moves in its linear system. Inparticular, Θ null is a uniruled divisor for g ≤ .We determine the numerical parameters of the family Γ ⊂ S + g induced by varying C ⊂ S . Since C = 0 , the pencil | C | is base point free and gives rise to a fibration f : ˜ S → P , where ˜ S := Bl ( S ) is the blow-up of the nodes of S . This in turn induces amoduli map m : P → S + g and Γ =: m ( P ) . We have the formulas Γ · λ = m ∗ ( λ ) = χ ( S, O S ) + g − g − g + 7 , ODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES 19 and Γ · α + 2Γ · β = m ∗ ( π ∗ ( δ )) = m ∗ ( α ) + 2 m ∗ ( β ) = c ( ˜ S ) + 4( g − . Noether’s formula gives that c ( ˜ S ) = 12 χ ( ˜ S, O ˜ S ) − K S = 12 χ ( S, O S ) − K S = 80 , hence m ∗ ( α ) + 2 m ∗ ( β ) = 4 g + 76 . The singular fibres corresponding to spin curves lying in B are those in the fibres over the blown-up nodes and all contribute with multiplicity ,that is, Γ · β = 8 and then Γ · α = 4 g + 60 . It follows that Γ · Θ null = − < (independentof g !), which finishes the proof. (ii) g = 5 . In the case C ⊂ Q ⊂ P and we choose a general quartic X ∈ H ( P , I C/ P (4)) and set S := Q ∩ X . Then S is a canonical surface with nodes at the points X ∩ Sing( Q ) .As in the previous case dim | C | = 1 , and the numerical characters of the induced family Γ ⊂ S +5 can be readily computed: Γ · λ = g + 5 = 10 , Γ · β = | Sing( S ) | = 4 , and Γ · α = 4 g + 52 , where the last equality is a consequence of Noether’s formula Γ · ( α +2 β ) = 12 χ ( S, O S ) − K S +4( g −
1) = 4 g +60 . By direct calculation, we obtain once more that Γ · Θ null = − . Thecase g = 6 is similar, except that the canonical surface S is a (2 , , complete intersectionin P , where one of the quadrics is the rank quadric Q . (iii) g = 4 . In this last case we proceed slightly differently and denote by S = F the blow-up of the vertex of a cone Q ⊂ P over a conic in P and write Pic( S ) = Z · F + Z · C , where F = 0 , C = − and C · F = 1 . We choose a Lefschetz pencil of genus curves in thelinear system | C + 2 F ) | . By blowing-up the
18 = 9( C + 2 F ) base points, we obtaina fibration f : ˜ S := Bl ( S ) → P which induces a family of spin curves m : P → S +4 given by m ( t ) := [ f − ( t ) , O f − ( t ) ( F )] . We have the formulas m ∗ ( λ ) = χ ( ˜ S, O ˜ S ) + g − , and m ∗ ( π ∗ ( δ )) = m ∗ ( α ) + 2 m ∗ ( β ) = c ( ˜ S ) + 4( g −
1) = 34 . The singular fibres lying in B correspond to curves in the Lefschetz pencil on Q passingthrough the vertex of the cone, that is, when f − ( t ) splits as C + D , where D ⊂ ˜ S is theresidual curve. Since C · D = 2 and O C ( F ) = O C (1) , it follows that m ( t ) ∈ B . Onefinds that m ∗ ( β ) = 1 , hence m ∗ ( α ) = 32 and m ∗ (Θ null ) = − . Since Γ := m ( P ) fills-upthe divisor Θ null , we obtain that [Θ null ] ∈ Eff( S +4 ) is rigid. (cid:3)
5. S
PIN CURVES OF GENUS The moduli space M carries one Brill-Noether divisor, the locus of plane septics M , := { [ C ] ∈ M : G ( C ) = ∅} . The locus M , is irreducible and for a known constant c , ∈ Z > , one has, cf. [EH1], bn := 1 c , M , ≡ λ − δ − δ − δ − δ − δ ∈ Pic ( M ) . In particular, s ( M , ) = 6 + 12 / ( g + 1) and this is the minimal slope of an effective divisoron M . The following fact is probably well-known: Proposition 5.1.
Through a general point of M , there passes a rational curve R ⊂ M suchthat R · M , < . In particular, the class [ M , ] ∈ Eff( M ) is rigid. Proof.
One takes a Lefschetz pencil of nodal plane septic curves with assigned nodes ingeneral position (and unassigned base points). After blowing up the unassignedbase points as well as the nodes, we obtain a fibration f : S := Bl ( P ) → P , and thecorresponding moduli map m : P → M is a covering curve for the irreducible divisor M , . The numerical invariants of this pencil are m ∗ ( λ ) = χ ( S, O S ) + g − and m ∗ ( δ ) = c ( S ) + 4( g −
1) = 59 , while m ∗ ( δ i ) = 0 for i = 1 , . . . , . We find m ∗ ([ M , ]) = c , (8 · − ·
59) = − c , < . (cid:3) Using (5) we find the following explicit representative for the canonical class K S +8 :(6) K S +8 ≡ π ∗ ( bn ) + 8Θ null + X i =1 ( a i α i + b i β i ) , where a i , b i > for i = 1 , . . . , . The multiples of each irreducible component appearingin (6) are rigid divisors on S +8 , but in principle, their sum could still be a movable class.Assuming for a moment Proposition 0.9, we explain how this implies Theorem 0.1: Proof of Theorem 0.1.
The covering curve R ⊂ Θ null constructed in Proposition 0.9, satisfies R · Θ null < as well as R · π ∗ ( M , ) = 0 and R · α i = R · β i = 0 for i = 1 , . . . , . It followsfrom (6) that for each n ≥ , one has an equality of linear series on S +8 | nK S +8 | = 8 n Θ null + | n ( K S +8 − null ) | . Furthermore, from (6) one finds constants a ′ i > for i = 1 , . . . , , such that if D ≡ λ − δ − X i =1 a ′ i δ i ∈ Pic ( M ) , then the difference π ∗ ( D ) − ( K S +8 − null ) is still effective on S +8 . We can thus write ≤ κ ( S +8 ) = κ (cid:0) S +8 , K S +8 − null (cid:1) ≤ κ (cid:0) S +8 , π ∗ ( D ) (cid:1) = κ (cid:0) S +8 , π ∗ ( D ) (cid:1) . We claim that κ (cid:0) S +8 , π ∗ ( D ) (cid:1) = 0 . Indeed, in the course of the proof of Proposition 5.1 wehave constructed a covering family B ⊂ M for the divisor M , such that B · M , < and B · δ i = 0 for i = 1 , . . . , . We lift B to a family R ⊂ S +8 of spin curves by taking ˜ B := B × M S +8 = { [ C t , η C t ] ∈ S +8 : [ C t ] ∈ B, η C t ∈ Pic ( C t ) , t ∈ P } ⊂ S +8 . One notes that ˜ B is disjoint from the boundary divisors A i , B i ⊂ S +8 for i = 1 , . . . , ,hence ˜ B · π ∗ ( D ) = 2 g − (2 g + 1)( B · M , ) M < . Thus we write that κ (cid:0) S +8 , π ∗ ( D ) (cid:1) = κ (cid:0) S +8 , π ∗ ( D − (22 λ − δ ) (cid:1) = κ (cid:0) S +8 , X i =1 a ′ i ( α i + β i ) (cid:1) = 0 . (cid:3) ODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES 21
6. A
FAMILY OF SPIN CURVES R ⊂ S +8 WITH R · π ∗ ( M , ) = 0 AND R · Θ null < The aim of this section is to prove Proposition 0.9, which is the key ingredient inthe proof of Theorem 0.1. We begin by reviewing facts about the geometry of M , inparticular the construction of general curves of genus 8 as complete intersections in arational homogeneous variety, see [M2].We fix V := C and denote by G := G (2 , V ) ⊂ P ( ∧ V ) the Grassmannian of lines.Noting that smooth codimension linear sections of G are canonical curves of genus ,one is led to consider the Mukai model of the moduli space of curves of genus M := G (8 , ∧ V ) st //SL ( V ) . There is a birational map f : M M , whose inverse is given by f − ( H ) := G ∩ H ,for a general H ∈ G (8 , ∧ V ) . The map f is constructed as follows: Starting with a curve [ C ] ∈ M − M , , one notes that C has a finite number of pencils g . We choose A ∈ W ( C ) and set L := K C ⊗ A ∨ ∈ W ( C ) . There exists a unique rank vector bundle E ∈ SU C (2 , K C ) (independent of A !), sitting in an extension −→ A −→ E −→ L −→ , such that h ( E ) = h ( A ) + h ( L ) = 6 . Since E is globally generated, we define the map φ E : C → G (cid:0) , H ( E ) ∨ (cid:1) , φ E ( p ) := E ( p ) ∨ (cid:0) ֒ → H ( E ) ∨ (cid:1) , and let ℘ : G (2 , H ( E ) ∨ ) → P ( ∧ H ( E ) ∨ ) be the Pl ¨ucker embedding. The determinantmap u : ∧ H ( E ) → H ( K C ) is surjective and we can view H ( K C ) ∨ ∈ G (8 , ∧ H ( E ) ∨ ) ,see [M2] Theorem C. We set f ([ C ]) := H ( K C ) ∨ mod SL ( H ( E ) ∨ ) ∈ M , that is, we assign to C its linear span h C i under the Pl ¨ucker map ℘ ◦ φ E : C → P (cid:0) ∧ H ( E ) ∨ (cid:1) .Even though this is not strictly needed for our proof, it follows from [M2] thatthe exceptional divisors of f are the Brill-Noether locus M , and the boundary divisors ∆ , . . . , ∆ . The map f − does not contract any divisors.Inside the moduli space F of polarized K surfaces [ S, h ] of degree h = 14 , weconsider the following Noether-Lefschetz divisor NL := { [ S, O S ( C + C )] ∈ F : Pic( S ) ⊃ Z · C ⊕ Z · C , C = C = 0 , C · C = 7 } , of doubly-elliptic K surfaces. For a general element [ S, O S ( C )] ∈ NL , the embeddedsurface φ O S ( C ) : S ֒ → P lies on a rank quadric whose rulings induce the elliptic pencils | C | and | C | on S .Let U → NL be the space classifying pairs (cid:0) [ S, O S ( C + C )] , C ⊂ S (cid:1) , where C ∈ | H ( S, O S ( C )) ⊗ H ( S, O S ( C )) | ⊂ | H ( S, O S ( C + C )) | . An element of U corresponds to a hyperplane section C ⊂ S ⊂ P of a doubly-elliptic K surface, such that the intersection of h C i with the rank quadric induced by the ellipticpencils, has rank at most . There exists a rational map q : U Θ null , q (cid:0) [ S, O S ( C + C )] , C (cid:1) := [ C, O C ( C ) = O C ( C )] . Since U is birational to a P -bundle over an open subvariety of NL , we obtain that U isirreducible and dim ( U ) = 21 (cid:0) = 3 + dim ( NL ) (cid:1) . We shall show that the morphism q isdominant (see Corollary 6.3) and begin with some preparations. We fix a general point [ C, η ] ∈ Θ null ⊂ S +8 , with η a vanishing theta-null. Then C ⊂ Q ⊂ P := P (cid:0) H ( C, K C ) ∨ (cid:1) , where Q ∈ H ( P , I C/ P (2)) is the rank quadric such that the ruling of Q cuts out on C precisely η . As explained, there exists a linear embedding P ⊂ P := P (cid:0) ∧ H ( E ) ∨ (cid:1) such that P ∩ G = C . The restriction map yields an isomorphism between spaces ofquadrics, cf. [M2], res C : H ( G , I G / P (2)) ∼ = −→ H ( P , I C/ P (2)) . In particular there is a unique quadric G ⊂ ˜ Q ⊂ P such that ˜ Q ∩ P = Q .There are three possibilities for the rank of any quadric ˜ Q ∈ H ( P , I G / P (2)) : (a) rk( ˜ Q ) = 15 , (b) rk( ˜ Q ) = 6 and then ˜ Q is a Pl ¨ucker quadric , or (c) rk ( ˜ Q ) = 10 , in whichcase ˜ Q is a sum of two Pl ¨ucker quadrics, see [M2] Proposition 1.4. Proposition 6.1.
For a general [ C, η ] ∈ Θ null , the quadric ˜ Q is smooth, that is, rk( ˜ Q ) = 15 .Proof. We may assume that dim G ( C ) = 0 (in particular C has no g ’s), and G ( C ) = ∅ .The space P (Ker( u )) ⊂ P (cid:0) ∧ H ( E ) (cid:1) is identified with the space of hyperplanes H ∈ ( P ) ∨ containing the canonical space P . Claim:
If rk ( ˜ Q ) < , there exists a pencil of -dimensional planes P ⊂ Ξ ⊂ P , suchthat S := G ∩ Ξ is a K surface containing C as a hyperplane section, and rk (cid:8) Q Ξ := ˜ Q ∩ Ξ ∈ H (Ξ , I S/ Ξ (2)) (cid:9) = 3 . The conclusion of the claim contradicts the assumption that [ C, η ] ∈ Θ null is general.Indeed, we pick such an -plane Ξ and corresponding K surface S . Since Sing ( Q ) ∩ C = ∅ , where Q Ξ ∩ P = Q , it follows that S ∩ Sing ( Q Ξ ) is finite. The ruling of Q Ξ cuts outan elliptic pencil | E | on S . Furthermore, S has nodes at the points S ∩ Sing ( Q Ξ ) . Fornumerical reasons, | Sing( S ) | = 7 , and then on the surface ˜ S obtained from S by resolvingthe nodes, one has the linear equivalence C ≡ E + Γ + · · · + Γ , where Γ i = − , Γ i · E = 1 for i = 1 , . . . , and Γ i · Γ j = 0 for i = j . In particularrk ( Pic ( ˜ S )) ≥ . A standard parameter count, see e.g. [Do1], shows that dim (cid:8) ( S, C ) : C ∈ |O S (2 E + Γ + · · · + Γ ) | (cid:9) ≤ − dim |O ˜ S ( C ) | = 20 . Since dim (Θ null ) = 20 and a general curve [ C ] ∈ Θ null lies on infinitely many such K surfaces S , one obtains a contradiction.We are left with proving the claim made in the course of the proof. The key pointis to describe the intersection P ( Ker ( u )) ∩ ˜ Q ∨ , where we recall that the linear span h ˜ Q ∨ i classifies hyperplanes H ∈ ( P ) ∨ such that rk ( ˜ Q ∩ H ) ≤ rk ( ˜ Q ) − . Note also thatdim h ˜ Q i = rk ( ˜ Q ) − .If rk ( ˜ Q ) = 6 , then ˜ Q ∨ is contained in the dual Grassmannian G ∨ := G (2 , H ( E )) ,cf. [M2] Proposition 1.8. Points in the intersection P ( Ker ( u )) ∩ G ∨ correspond to decom-posable tensors s ∧ s , with s , s ∈ H ( C, E ) , such that u ( s ∧ s ) = 0 . The image of themorphism O ⊕ C ( s ,s ) −→ E is thus a subbundle g of E and there is a bijection P (Ker( u )) ∩ G (cid:0) , H ( E ) (cid:1) ∼ = W ( C ) . ODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES 23
It follows, there are at most finitely many tangent hyperplanes to ˜ Q containing the space P = h C i , and consequently, dim (cid:0) P ( Ker ( u )) ∩ h ˜ Q ∨ i (cid:1) ≤ . Then there exists a codimension linear space W ⊂ P such that rk ( ˜ Q ∩ W ) = 3 , which proves the claim (and muchmore), in the case rk ( ˜ Q ) = 6 .When rk ( ˜ Q ) = 10 , using the explicit description of the dual quadric ˜ Q ∨ providedin [M2] Proposition 1.8, one finds that dim (cid:0) P ( Ker ( u )) ∩ h ˜ Q ∨ i (cid:1) ≤ . Thus there exists acodimension linear section W ⊂ P such that rk ( ˜ Q ∩ W ) = 3 , which implies the claimwhen rk ( ˜ Q ) = 10 as well. (cid:3) We consider an -dimensional linear extension P ⊂ Λ ⊂ P of the canonicalspace P = h C i , such that S Λ := Λ ∩ G is a smooth K3 surface. The restriction map res C/S Λ : H (Λ , I S Λ / Λ (2)) → H ( P , I C/ P (2)) is an isomorphism, see [SD]. Thus there exists a unique quadric S Λ ⊂ Q Λ ⊂ Λ with Q Λ ∩ P = Q . Since rk ( Q ) = 3 , it follows that ≤ rk ( Q Λ ) ≤ and it is easy to see that fora general Λ , the corresponding quadric Q Λ ⊂ Λ is of rank . We show however, that onecan find K -extensions of the canonical curve C , which lie on quadrics of rank : Proposition 6.2.
For a general [ C, η ] ∈ Θ null , there exists a pencil of -dimensional extensions P ( H ( C, K C ) ∨ ) ⊂ Λ ⊂ P such that rk( Q Λ ) = 4 . It follows that there exists a smooth K surface S Λ ⊂ Λ containing C asa transversal hyperplane section, such that rk( Q Λ ) = 4 .Proof. We pass from projective to vector spaces and view the rank quadric ˜ Q : ∧ H ( C, E ) ∨ ∼ −→ ∧ H ( C, E ) as an isomorphism, which by restriction to H ( C, K C ) ∨ ⊂ ∧ H ( C, E ) ∨ , induces therank quadric Q : H ( C, K C ) ∨ → H ( C, K C ) . The map u ◦ ˜ Q : ∧ H ( E ) ∨ → H ( K C ) being surjective, its kernel Ker ( u ◦ ˜ Q ) is a -dimensional vector space containing the -dimensional subspace Ker ( Q ) . We choose an arbitrary element [¯ v := v + Ker( Q )] ∈ P (cid:16) Ker( u ◦ ˜ Q )Ker( Q ) (cid:17) = P , inducing a subspace H ( C, K C ) ∨ ⊂ Λ := H ( C, K C ) ∨ + C v ⊂ ∧ H ( C, E ) ∨ , with theproperty that Ker ( Q Λ ) = Ker ( Q ) , where Q Λ : Λ → Λ ∨ is induced from ˜ Q by restrictionand projection. It follows that rk ( Q Λ ) = 4 and there is a pencil of -planes Λ ⊃ P withthis property. (cid:3) Let C ⊂ Q ⊂ P be a general canonical curve endowed with a vanishing theta-null,where Q ∈ H (cid:0) P , I C/ P (2) (cid:1) is the corresponding rank quadric. We choose a general -plane P ⊂ Λ ⊂ P such that S := Λ ∩ G is a smooth K3 surface, and the lift of Q to Λ Q Λ ∈ H (cid:0) Λ , I S/ Λ (2) (cid:1) has rank (cf. Proposition 6.2). Moreover, we can assume that S ∩ Sing ( Q Λ ) = ∅ . Thelinear projection f Λ : Λ P with center Sing ( Q Λ ) , induces a regular map f : S → P with image the smooth quadric Q ⊂ P . Then S is endowed with two elliptic pencils | C | and | C | corresponding to the projections of Q ∼ = P × P onto the two factors. Since C ∈ |O S (1) | , one has a linear equivalence C ≡ C + C , on S . As already pointed out,deg ( f ) = C · C = C / . The condition rk ( Q Λ ∩ P ) = rk ( Q Λ ) − , implies that thehyperplane P ∈ (Λ) ∨ is the pull-back of a hyperplane from P , that is, P = f − (Π ) ,where Π ∈ ( P ) ∨ . This proves the following: Corollary 6.3.
The rational morphism q : U Θ null is dominant.Proof. Keeping the notation from above, if [ C ] ∈ Θ null is a general point corresponding tothe rank quadric Q ∈ H ( P , I C/ P (2)) , then [ S, O S ( C + C ) , C ] ∈ q − ([ C ]) . (cid:3) We begin the proof of Proposition 0.9 while retaining the set-up described above.Let us choose a general line l ⊂ Π and denote by { q , q } := l ∩ Q . We consider thepencil { Π t } t ∈ P ⊂ ( P ) ∨ of planes through l as well as the induced pencil of curves ofgenus { C t := f − (Π t ) ⊂ S } t ∈ P , each endowed with a vanishing theta-null induced by the pencil f t : C t → Q ∩ Π t .This pencil contains precisely two reducible curves, corresponding to the planes Π , Π in P spanned by the rulings of Q passing through q and q respectively. Pre-cisely, if l i , m i ⊂ Q are the rulings passing through q i such that l · l = m · m = 0 , then itfollows that for Π = h l , m i , Π = h l , m i , the fibres f − (Π ) and f − (Π ) split into twoelliptic curves f − ( l i ) and f − ( m j ) meeting transversally in points. The half-canonical g specializes to a degree admissible covering f − ( l i ) ∪ f − ( m j ) f → l i ∪ m j , i = j, such that the points in f − ( l i ) ∩ f − ( m j ) map to l i ∩ m j . To determine the point in S +8 corresponding to the admissible covering (cid:0) f − ( l i ) ∪ f − ( m j ) , f | f − ( l i ) ∪ f − ( m j ) (cid:1) , one mustinsert exceptional components at all the points of intersection of the two components.We denote by R ⊂ Θ null ⊂ S +8 the pencil of spin curves obtained via this construction. Lemma 6.4.
Each member C t ⊂ S in the above constructed pencil is nodal. Moreover, eachcurve C t different from f − ( l ) ∪ f − ( m ) and f − ( l ) ∪ f − ( m ) is irreducible. It follows that R · α i = R · β i = 0 for i = 1 , . . . , .Proof. This follows since f : S → Q is a regular morphism and the base line l ⊂ H ofthe pencil { Π t } t ∈ P is chosen to be general. (cid:3) Lemma 6.5. R · π ∗ ( M , ) = 0 .Proof. We show instead that π ∗ ( R ) · M , = 0 . From Lemma 6.4, the curve R is disjointfrom the divisors A i , B i for i = 1 , . . . , , hence π ∗ ( R ) has the numerical characteristics ofa Lefschetz pencil of curves of genus on a fixed K surface.In particular, π ∗ ( R ) · δ/π ∗ ( R ) · λ = 6 + 12 / ( g + 1) = s ( M , ) and π ∗ ( R ) · δ i = 0 for i = 1 , . . . , . This implies the statement. (cid:3) Lemma 6.6. R · Θ null = − . ODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES 25
Proof.
We have already determined that R · λ = π ∗ ( R ) · λ = χ ( ˜ S, O ˜ S ) + g − , where ˜ S := Bl g − ( S ) is the blow-up of S at the points f − ( q ) ∪ f − ( q ) . Moreover,(7) R · α + 2 R · β = π ∗ ( R ) · δ = c ( ˜ X ) + 4( g −
1) = 38 + 28 = 66 . To determine R · β we study the local structure of S +8 in a neighbourhood of one of thetwo points, say t ∗ ∈ R corresponding to a reducible curve, say f − ( l ) ∪ f − ( m ) , thesituation for f − ( l ) ∪ f − ( m ) being of course identical. We set { p } := l ∩ m ∈ Q and { x , . . . , x } := f − ( p ) ⊂ S . We insert exceptional components E , . . . , E at the nodes x , . . . , x of f − ( l ) ∪ f − ( m ) and denote by X the resulting quasi-stable curve. If µ : f − ( l ) ∪ f − ( m ) ∪ E ∪ . . . ∪ E → f − ( l ) ∪ f − ( m ) is the stabilization morphism, we set { y i , z i } := µ − ( x i ) , where y i ∈ E i ∩ f − ( l ) and z i ∈ E i ∩ f − ( m ) for i = 1 , . . . , . If t ∗ = [ X, η, β ] , then η f − ( l ) = O f − ( l ) , η f − ( m ) = O f − ( m ) , and of course η E i = O E i (1) . Moreover, one computes that Aut ( X, η, β ) = Z ,see [Cor] Lemma 2.2, while clearly Aut ( f − ( l ) ∪ f − ( m )) = { Id } .If C g − τ denotes the versal deformation space of [ X, η, β ] ∈ S + g , then there arelocal parameters ( τ , . . . , τ g − ) , such that for i = 1 , . . . , , the locus (cid:0) τ i = 0 (cid:1) ⊂ C g − τ parameterizes spin curves for which the exceptional component E i persists. It particular,the pull-back C g − τ × S + g B of the boundary divisor B ⊂ S + g is given by the equation (cid:0) τ · · · τ = 0 (cid:1) ⊂ C g − τ . The group Aut ( X, η, β ) acts on C g − τ by ( τ , . . . , τ , τ , . . . , τ g − ) ( − τ , . . . , − τ , τ , . . . , τ g − ) , and since an ´etale neighbourhood of t ∗ ∈ S + g is isomorphic to C g − τ / Aut ( X, η, β ) , wefind that the Weil divisor B is not Cartier around t ∗ (though B is Cartier). It followsthat the intersection multiplicity of R × S + g C g − τ with the locus ( τ · · · τ ) = 0 equals ,that is, the intersection multiplicity of R ∩ β at the point t ∗ equals / , hence R · β = (cid:0) R · β (cid:1) f − ( l ) ∪ f − ( m ) + (cid:0) R · β (cid:1) f − ( l ) ∩ f − ( m ) = 72 + 72 = 7 . Then using (7) we find that R · α = 66 −
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