The birational type of the moduli space of even spin curves
aa r X i v : . [ m a t h . AG ] A ug THE BIRATIONAL TYPE OF THE MODULI SPACE OF EVEN SPIN CURVES
GAVRIL FARKAS
The moduli space S g of smooth spin curves parameterizes pairs [ C, η ] , where [ C ] ∈ M g is a curve of genus g and η ∈ Pic g − ( C ) is a theta-characteristic. The finiteforgetful map π : S g → M g has degree g and S g is a disjoint union of two connectedcomponents S + g and S − g of relative degrees g − (2 g + 1) and g − (2 g − correspondingto even and odd theta-characteristics respectively. A compactification S g of S g over M g is obtained by considering the coarse moduli space of the stack of stable spin curves ofgenus g (cf. [C], [CCC] and [AJ]). The projection S g → M g extends to a finite branchedcovering π : S g → M g . In this paper we determine the Kodaira dimension of S + g : Theorem 0.1.
The moduli space S + g of even spin curves is a variety of general type for g > and it is uniruled for g < . The Kodaira dimension of S +8 is non-negative . It was classically known that S +2 is rational. The Scorza map establishes a bira-tional isomorphism between S +3 and M , cf. [DK], hence S +3 is rational. Very recently,Takagi and Zucconi [TZ] showed that S +4 is rational as well. Theorem 0.1 can be com-pared to [FL] Theorem 0.3: The moduli space R g of Prym varieties of dimension g − (that is, non-trivial square roots of O C for each [ C ] ∈ M g ) is of general type when g > and g = 15 . On the other hand R g is unirational for g < . Surprisingly, the problemof determining the Kodaira dimension has a much shorter solution for S + g than for R g and our results are complete.We describe the strategy to prove that S + g is of general type for a given g . Wedenote by λ = π ∗ ( λ ) ∈ Pic ( S + g ) the pull-back of the Hodge class and by α , β ∈ Pic ( S + g ) and α i , β i ∈ Pic ( S + g ) for ≤ i ≤ [ g/ boundary divisor classes such that π ∗ ( δ ) = α + 2 β and π ∗ ( δ i ) = α i + β i for ≤ i ≤ [ g/ (see Section 2 for precise definitions). Using Riemann-Hurwitz and [HM] we find that K S + g ≡ π ∗ ( K M g ) + β ≡ λ − α − β − [ g/ X i =1 ( α i + β i ) − ( α + β ) . We prove that K S + g is a big Q -divisor class by comparing it against the class of theclosure in S + g of the divisor Θ null on S + g of non-vanishing even theta characteristics: Research partially supported by an Alfred P. Sloan Fellowship. Building on the results of this paper, we have proved quite recently in joint work with A. Verra, that κ ( S +8 ) = 0 . Details will appear later. heorem 0.2. The closure in S + g of the divisor Θ null := { [ C, η ] ∈ S + g : H ( C, η ) = 0 } ofnon-vanishing even theta characteristics has class equal to Θ null ≡ λ − α − [ g/ X i =1 β i ∈ Pic( S + g ) . Note that the coefficients of β and α i for ≤ i ≤ [ g/ in the expansion of [Θ null ] are equal to . To prove Theorem 0.2, one can use test curves on S + g or alternatively,realize Θ null as the push-forward of the degeneracy locus of a map of vector bundlesof the same rank defined over a certain Hurwitz scheme covering S + g and use [F1] and[F2] to compute the class of this locus. Then we use [FP] Theorem 1.1, to construct foreach genus ≤ g ≤ an effective divisor class D ≡ aλ − P [ g/ i =0 b i δ i ∈ Eff ( M g ) withcoefficients satisfying the inequalities ab ≤ g +1 , if g + 1 is composite , if g = 10 k + k − k ( k − , if g = 2 k − ≥ and b i /b ≥ / for ≤ i ≤ [ g/ . When g + 1 is composite we choose for D the closureof the Brill-Noether divisor of curves with a g rd , that is, M rg,d := { [ C ] ∈ M g : G rd ( C ) = ∅} in case when the Brill-Noether number ρ ( g, r, d ) = − , and then cf. [EH2] M rg,d ≡ c g,d,r (cid:16) ( g + 3) λ − g + 16 δ − [ g/ X i =1 i ( g − i ) δ i (cid:17) ∈ Pic( M g ) . For g = 10 we take the closure of the divisor K := { [ C ] ∈ M : C lies on a K surface } (cf. [FP] Theorem 1.6). In the remaining cases, when necessarily g = 2 k − , we choosefor D the Gieseker-Petri divisor GP g,k consisting of those curves [ C ] ∈ M g such thatthere exists a pencil A ∈ W k ( C ) such that the multiplication map µ ( A ) : H ( C, A ) ⊗ H ( C, K C ⊗ A ∨ ) → H ( C, K C ) is not an isomorphism, see [EH2], [F2]. Having chosen D , we form the Q -linear combi-nation of divisor classes · Θ null + 32 b · π ∗ ( D ) = (cid:0) a b (cid:1) λ − α − β − [ g/ X i =1 b i b α i − [ g/ X i =1 (cid:0) b i (cid:1) β i ∈ Pic( S + g ) , from which we can write K S + g = ν g · λ + 8Θ null + 32 b π ∗ ( D ) + [ g/ X i =1 (cid:0) c i · α i + c ′ i · β i ) , where c i , c ′ i ≥ . Moreover ν g > precisely when g ≥ , while ν = 0 . Since the class λ ∈ Pic ( S + g ) is big and nef, we obtain that K S + g is a big Q -divisor class on the normalvariety S + g as soon as g > . It is proved in [Lud] that for g ≥ pluricanonical formsdefined on S + g, reg extend to any resolution of singularities c S + g → S + g , which shows that S + g is of general type whenever ν g > and completes the proof of Theorem 0.1 for g ≥ . hen g ≤ we show that K S + g / ∈ Eff ( S + g ) by constructing a covering curve R ⊂ S + g such that R · K S + g < , cf. Theorem 1.2. We then use [BDPP] to conclude that S + g isuniruled.I would like to thank the referee for pertinent comments which led to a clearlyimproved version of this paper.1. T HE STACK OF SPIN CURVES
We review a few facts about Cornalba’s compactification π : S g → M g , see [C].If X is a nodal curve, a smooth rational component E ⊂ X is said to be exceptional if E ∩ X − E ) = 2 . The curve X is said to be quasi-stable if E ∩ X − E ) ≥ for anysmooth rational component E ⊂ X , and moreover any two exceptional components of X are disjoint. A quasi-stable curve is obtained from a stable curve by blowing-up eachnode at most once. We denote by [ st ( X )] ∈ M g the stable model of X . Definition 1.1. A spin curve of genus g consists of a triple ( X, η, β ) , where X is a genus g quasi-stable curve, η ∈ Pic g − ( X ) is a line bundle of degree g − such that η E = O E (1) for every exceptional component E ⊂ X , and β : η ⊗ → ω X is a sheaf homomorphismwhich is generically non-zero along each non-exceptional component of X .A family of spin curves over a base scheme S consists of a triple ( X f → S, η, β ) , where f : X → S is a flat family of quasi-stable curves, η ∈ Pic( X ) is a line bundle and β : η ⊗ → ω X is a sheaf homomorphism, such that for every point s ∈ S the restriction ( X s , η X s , β X s : η ⊗ X s → ω X s ) is a spin curve.To describe locally the map π : S g → M g we follow [C] Section 5. We fix [ X, η, β ] ∈ S g and set C := st ( X ) . We denote by E , . . . , E r the exceptional componentsof X and by p , . . . , p r ∈ C sing the nodes which are images of exceptional components.The automorphism group of ( X, η, β ) fits in the exact sequence of groups −→ Aut ( X, η, β ) −→ Aut ( X, η, β ) res C −→ Aut ( C ) . We denote by C g − τ the versal deformation space of ( X, η, β ) where for ≤ i ≤ r thelocus ( τ i = 0) ⊂ C g − τ corresponds to spin curves in which the component E i ⊂ X persists. Similarly, we denote by C g − t = Ext (Ω C , O C ) the versal deformation spaceof C and denote by ( t i = 0) ⊂ C g − t the locus where the node p i ∈ C is not smoothed.Then around the point [ X, η, β ] , the morphism π : S g → M g is locally given by the map(1) C g − τ Aut ( X, η, β ) → C g − t Aut ( C ) , t i = τ i (1 ≤ i ≤ r ) and t i = τ i ( r + 1 ≤ i ≤ g − . From now on we specialize to the case of even spin curves and describe the boundaryof S + g . In the process we determine the ramification of the finite covering π : S + g → M g .1.1. The boundary divisors of S + g . If [ X, η, β ] ∈ π − ([ C ∪ y D ]) where [ C, y ] ∈ M i, and [ D, y ] ∈ M g − i, , then neces-sarily 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 ) , here 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 ofthe 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, .For a general point [ X, η, β ] ∈ A i ∪ B i we have that Aut ( X, η, β ) =
Aut ( X, η, β ) = Z . Using (1), the map C g − τ → C g − t is given by t = τ and t i = τ i for i ≥ .Furthermore, Aut ( X, η, β ) acts on C g − τ via ( τ , τ , . . . , τ g − ) ( − τ , τ , . . . , τ g − ) .It follows that ∆ i ⊂ M g is not a branch divisor for π : S + g → M g and if α i = [ A i ] ∈ Pic( S + g ) and β i = [ B i ] ∈ Pic( S + g ) , then for ≤ i ≤ [ g/ we have the relation(2) π ∗ ( δ i ) = α i + β i . Moreover, π ∗ ( α i ) = 2 g − (2 i + 1)(2 g − i + 1) δ i and π ∗ ( β i ) = 2 g − (2 i − g − i − δ i .For a point [ X, η, β ] such that st ( X ) = C yq := C/y ∼ q , with [ C, y, q ] ∈ M g − , ,there are two possibilities depending on whether X possesses an exceptional compo-nent or not. If X = C yq and η C := ν ∗ ( η ) where ν : C → X denotes the normalizationmap, then η ⊗ C = K C ( y + q ) . For each choice of η C ∈ Pic g − ( C ) as above, there is pre-cisely one choice of gluing the fibres η C ( y ) and η C ( q ) such that h ( X, η ) ≡ mod . Wedenote by A the closure in S + g of the locus of points [ C yq , η C ∈ p K C ( y + q )] as aboveand clearly deg ( A / ∆ ) = 2 g − .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 − .For [ C, y, q ] ∈ M g − , sufficiently generic we have that Aut ( X, η, β ) =
Aut ( C ) = { Id C } ,and then from (1) it follows that π is simply branched over such points. We denote by B ⊂ S + g the closure of the locus of points [ C ∪ { y,q } E, η C ∈ √ K C , η E = O E (1)] . If α = [ A ] ∈ Pic ( S + g ) and β = [ B ] ∈ Pic ( S + g ) , we then have the relation(3) π ∗ ( δ ) = α + 2 β . Note that π ∗ ( α ) = 2 g − δ and π ∗ ( β ) = 2 g − (2 g − + 1) δ .1.2. The uniruledness of S + g for small g . We employ a simple negativity argument to determine κ ( S + g ) for small genus.Using an analogous idea we showed that similarly, for the moduli space of Prym curves,one has that κ ( R g ) = −∞ for g < , cf. [FL] Theorem 0.7. Theorem 1.2.
For g < , the space S + g is uniruled.Proof. We start with a fixed K surface S carrying a Lefschetz pencil of curves of genus g . This induces a fibration f : Bl g ( S ) → P and then we set B := (cid:0) m f (cid:1) ∗ ( P ) ⊂ M g ,where m f : P → M g is the moduli map m f ( t ) := [ f − ( t )] . We have the followingwell-known formulas on M g (cf. [FP] Lemma 2.4): B · λ = g + 1 , B · δ = 6 g + 18 , and B · δ i = 0 for i ≥ . We lift B to a pencil R ⊂ S + g of spin curves by taking R := B × M g S + g = { [ C t , η C t ] ∈ S + g : [ C t ] ∈ B, η C t ∈ Pic g − ( C t ) , t ∈ P } ⊂ S + g . sing (3) one computes the intersection numbers with the generators of Pic ( S + g ) : R · λ = ( g + 1)2 g − (2 g + 1) , R · α = (6 g + 18)2 g − and R · β = (6 g + 18)2 g − (2 g − + 1) . Furthermore, R is disjoint from all the remaining boundary classes of S + g , that is, R · α i = R · β i = 0 for ≤ i ≤ [ g/ . One verifies that R · K S + g < precisely when g ≤ . Since R is a covering curve for S + g in the range g ≤ , we find that K S + g is not pseudo-effective,that is, K S + g ∈ Eff ( S + g ) c . Pseudo-effectiveness of the canonical bundle is a birationalproperty for normal varieties, therefore the canonical bundle of any smooth model of S + g lies outside the pseudo-effective cone as well. One can apply [BDPP] Corollary 0.3,to conclude that S + g is uniruled for g ≤ . (cid:3)
2. T
HE GEOMETRY OF THE DIVISOR Θ null We compute the class of the divisor Θ null using test curves. The same calculationcan be carried out using techniques developed in [F1], [F2] to calculate push-forwardsof tautological classes from stacks of limit linear series g rd (see also Remark 2.1).For g ≥ , Harer [H] has showed that H ( S + g , Q ) ∼ = Q . The range for which thisresult holds has been recently improved to g ≥ in [P]. In particular, it follows thatPic ( S + g ) Q is generated by the classes λ , α i , β i for i = 0 , . . . , [ g/ . Thus we can expandthe divisor class Θ null in terms of the generators of the Picard group(4) Θ null ≡ ¯ λ · λ − ¯ α · α − ¯ β · β − [ g/ X i =1 (cid:0) ¯ α i · α i + ¯ β i · β i (cid:1) ∈ Pic( S + g ) Q , and determine the coefficients ¯ λ, ¯ α , ¯ β , ¯ α i and ¯ β i ∈ Q for ≤ i ≤ [ g/ . Remark 2.1.
To show that the class [Θ null ] ∈ Pic ( S + g ) Q is a multiple of λ and thus, theexpansion (4) makes sense for all g ≥ , one does not need to know that Pic ( S + g ) Q isinfinite cyclic. For instance, for even g = 2 k − ≥ , we note that, via the base pointfree pencil trick, [ C, η ] ∈ Θ null if and only if the multiplication map µ C ( A, η ) : H ( C, A ) ⊗ H ( C, A ⊗ η ) → H ( C, A ⊗ ⊗ η ) is not an isomorphism for a base point free pencil A ∈ W k ( C ) . We set f M g to be theopen subvariety consisting of curves [ C ] ∈ M g such that W k − ( C ) = ∅ and denote by σ : G k → f M g the Hurwitz scheme of pencils g k and by τ : G k × f M g S + g → S + g , u : G k × f M g S + g → G k the (generically finite) projections. Then Θ null = τ ∗ ( Z ) , where Z = { [ A, C, η ] ∈ G k × f M g S + g : µ C ( A, η ) is not injective } . Via this determinantal presentation, the class of the divisor Z is expressible as a combi-nation of τ ∗ ( λ ) , u ∗ ( a ) , u ∗ ( b ) , where a , b ∈ Pic( G k ) Q are the tautological classes definedin e.g. [FL] p.15. Since τ ∗ ( u ∗ ( a )) = π ∗ ( σ ∗ ( a )) (and similarly for the class b ), the conclu-sion follows. For odd genus g = 2 k − , one uses a similar argument replacing G k withany generically finite covering of M g given by a Hurwitz scheme (for instance, we takethe space of pencils g k +1 with a triple ramification point). e start the proof of Theorem 0.2 by determining the coefficients of α i and β i ( i ≥ in the expansion of [Θ null ] . Theorem 2.2.
We fix integers g ≥ and ≤ i ≤ [ g/ . The coefficient of α i in the expansionof [Θ null ] equals , while the coefficient of β i equals − / . That is, ¯ α i = 0 and ¯ β i = 1 / .Proof. For each integer ≤ i ≤ g − , we fix general curves [ C ] ∈ M i and [ D, q ] ∈ M g − i, and consider the test curve C i := { C ∪ y ∼ q D } y ∈ C ⊂ ∆ i ⊂ M g . We lift C i to testcurves F i ⊂ A i and G i ⊂ B i inside S + g constructed as follows. We fix even (resp. odd)theta-characteristics η + C ∈ Pic i − ( C ) and η + D ∈ Pic g − i − ( D ) (resp. η − C ∈ Pic i − ( C ) and η − D ∈ Pic g − i − ( D ) ).If E ∼ = P is an exceptional component, we define the family F i (resp. G i ) asconsisting of spin curves F i := (cid:8) t := [ C ∪ y E ∪ q D, η C = η + C , η E = O E (1) , η D = η + D ] ∈ S + g : y ∈ C (cid:9) and G i := (cid:8) t := [ C ∪ y E ∪ q D, η C = η − C , η E = O E (1) , η D = η − D ] ∈ S + g : y ∈ C (cid:9) . Since π ∗ ( F i ) = π ∗ ( G i ) = C i , clearly F i · α i = C i · δ i = 2 − i, F i · β i = 0 and F i hasintersection number with all other generators of Pic ( S + g ) . Similarly G i · β i = 2 − i, G i · α i = 0 , G i · λ = 0 , and G i does not intersect the remaining boundary classes in S + g .Next we determine F i ∩ Θ null . Assume that a point t ∈ F i lies in Θ null . Then thereexists a family of even spin curves ( f : X →
S, η, β ) , where S = Spec ( R ) , with R beinga discrete valuation ring and X is a smooth surface, such that, if , ξ ∈ S denote thespecial and the generic point of S respectively and X ξ is the generic fibre of f , then h ( X ξ , η ξ ) ≥ , h ( X ξ , η ξ ) ≡ mod 2 , η ⊗ ξ ∼ = ω X ξ and (cid:0) f − (0) , η f − (0) (cid:1) = t ∈ S + g . Following the procedure described in [EH1] p. 347-351, this data produces a limit linearseries g g − on C ∪ D , say l := (cid:16) l C = ( L C , V C ) , l D = ( L D , V D ) (cid:17) ∈ G g − ( C ) × G g − ( D ) , such that the underlying line bundles L C and L D respectively, are obtained from theline bundle ( η + C , η E , η + D ) by dropping the E -aspect and then tensoring the line bundles η + C and η + D by line bundles supported at the points y ∈ C and q ∈ D respectively. Fordegree reasons, it follows that L C = η + C ⊗ O C (( g − i ) y ) and L D = η + D ⊗ O D ( iq ) . Sinceboth C and D are general in their respective moduli spaces, we have that H ( C, η + C ) = 0 and H ( D, η + D ) = 0 . In particular a l C ( y ) ≤ g − i − and a l D ( q ) < a l D ( q ) ≤ i − ,hence a l C ( y ) + a l D ( q ) ≤ g − , which contradicts the definition of a limit g g − . Thus F i ∩ Θ null = ∅ . This implies that ¯ α i = 0 , for all ≤ i ≤ [ g/ (for i = 1 , one uses insteadthe curve F g − ⊂ A to reach the same conclusion).Assume that t ∈ G i ∩ Θ null . By the same argument as above, retaining also thenotation, there is an induced limit linear series on C ∪ D , ( l C , l D ) ∈ G g − ( C ) × G g − ( D ) , here L C = η − C ⊗ O C (( g − i ) y ) and L D = η − D ⊗ O D ( iq ) . Since [ C ] ∈ M i and [ D, q ] ∈M g − i, are both general, we may assume that h ( D, η − D ) = h ( C, η − C ) = 1 , q / ∈ supp ( η − D ) and that supp ( η − C ) consists of i − distinct points. In particular a l D ( q ) ≤ i , hence a l C ( y ) ≥ g − − a l D ( q ) ≥ g − i − . Since h ( C, η − C ) = 1 , it follows that one has in factequality, that is, a l C ( y ) = g − i − and then necessarily a l D ( q ) = i .Similarly, a l C ( y ) ≤ g − i + 1 (otherwise div ( η − C ) ≥ y , that is, supp ( η − C ) would benon-reduced, a contradiction), thus a l D ( q ) ≥ i − , and the last two inequalities must beequalities as well (one uses that h (cid:0) D, L D ⊗ O D ( − ( i − q ) (cid:1) = h ( D, η − D ⊗ O D ( q )) = 1 ,that is, a l D ( q ) < i − ). Since a l C ( y ) = g − i + 1 , we find that y ∈ supp ( η − C ) .To sum up, we have showed that ( l C , l D ) is a refined limit g g − and in fact(5) l D = | η − D ⊗O D (2 q ) | +( i − · q ∈ G g − ( D ) , l C = | η − C ⊗O C ( y ) | +( g − i − · y ∈ G g − ( C ) , hence a l D ( q ) = ( i − , i ) and a l C ( y ) = ( g − i − , g − i + 1) .To prove that the intersection between G i and Θ null is transversal, we followclosely [EH3] Lemma 3.4 (see especially the Remark on p. 45): The restriction Θ null | G i isisomorphic, as a scheme, to the variety τ : T g − ( G i ) → G i of limit linear series g g − onthe curves of compact type { C ∪ y ∼ q D : y ∈ C } , whose C and D -aspects are obtained bytwisting suitably at y ∈ C and q ∈ D the fixed theta-characteristics η − C and η − D respec-tively. Following the description of the scheme structure of this moduli space given in[EH1] Theorem 3.3 over an arbitrary base, we find that because G i consists entirely ofsingular spin curves of compact type, the scheme T g − ( G i ) splits as a product of thecorresponding moduli spaces of C and D -aspects respectively of the limits g g − . Bydirect calculation we have showed that T g − ( G i ) ∼ = supp( η − C ) × { l D } . Since supp( η − C ) is a reduced -dimensional scheme, we obtain that Θ null | G i is everywhere reduced. Itfollows that G i · Θ null = supp ( η − C ) = i − and then ¯ β i = ( G i · Θ null ) / (2 i − . Thisargument does not work for i = 1 , when one uses instead the intersection of Θ null with G g − , and this finishes the proof. (cid:3) Next we construct two pencils in S + g which are lifts of the standard degree pencil of elliptic tails in M g . We fix a general pointed curve [ C, q ] ∈ M g − , and a pencil f : Bl ( P ) → P of plane cubics together with a section σ : P → Bl ( P ) induced byone of the base points. We then consider the pencil R := { [ C ∪ q ∼ σ ( λ ) f − ( λ )] } λ ∈ P ⊂ M g .We fix an odd theta-characteristic η − C ∈ Pic g − ( C ) such that q / ∈ supp ( η − C ) and E ∼ = P will again denote an exceptional component. We define the family F := { [ C ∪ q E ∪ σ ( λ ) f − ( λ ) , η C = η − C , η E = O E (1) , η f − ( λ ) = O f − ( λ ) ] : λ ∈ P } ⊂ S + g . Since F ∩ A = ∅ , we find that F · β = π ∗ ( F ) · δ = − . Similarly, F · λ = π ∗ ( F ) · λ = 1 and obviously F · α i = F · β i = 0 for ≤ i ≤ [ g/ . For each of the points λ ∞ ∈ P corresponding to singular fibres of R , the associated η λ ∞ ∈ Pic g − ( C ∪ E ∪ f − ( λ ∞ )) are actual line bundles on C ∪ E ∪ f − ( λ ∞ ) (that is, we do not have to blow-up the extranode). Thus we obtain that F · β = 0 , therefore F · α = π ∗ ( F ) · δ = 12 . e also fix an even theta-characteristic η + C ∈ Pic g − ( C ) and consider the degree branched covering γ : S +1 , → M , forgetting the spin structure. We define the pencil G := { (cid:2) C ∪ q E ∪ σ ( λ ) f − ( λ ) , η C = η + C , η E = O E (1) , η f − λ ) ∈ γ − [ f − ( λ )] (cid:3) : λ ∈ P } ⊂ S + g . Since π ∗ ( G ) = 3 R , we have that G · λ = 3 . Obviously G · β = G · β = 0 , hence G · α = π ∗ ( G ) · δ = − . The map γ : S +1 , → M , is simply ramified over thepoint corresponding to j -invariant ∞ . Hence, G · α = 12 and G · β = 12 , which isconsistent with formula (3).The last pencil we construct lies in the boundary divisor B ⊂ S + g : Setting E ∼ = P for an exceptional component, we define H := { [ C ∪ { y,q } E, η C = η + C , η E = O E (1)] : y ∈ C } ⊂ S + g . The fibre of H over the point y = q ∈ C is the even spin curve (cid:2) C ∪ q E ′ ∪ q ′ E ′′ ∪ { q ′′ ,y ′′ } E, η C = η + C , η E ′ = O E ′ (1) , η E = O E (1) , η E ′′ = O E ′′ ( − (cid:3) , having as stable model [ C ∪ q E ∞ ] , where E ∞ := E ′′ /y ′′ ∼ q ′′ is the rational nodal curvecorresponding to j = ∞ . Here E ′ , E ′′ are rational curves, E ′ ∩ E ′′ = { q ′ } , E ∩ E ′′ = { q ′′ , y ′′ } and the stabilization map for C ∪ E ∪ E ′ ∪ E ′′ contracts the components E ′ and E , while identifying q ′′ and y ′′ .We find that H · λ = 0 , H · α i = H · β i = 0 for ≤ i ≤ [ g/ . Moreover H · α = 0 ,hence H · β = π ∗ ( H ) · δ = 1 − g . Finally, H · α = 1 and H · β = 0 . Theorem 2.3. If F , G , H ⊂ S + g are the families of spin curves defined above, then F · Θ null = G · Θ null = H · Θ null = 0 . Proof.
From the limit linear series argument in the proof of Theorem 2.2 we get thatthe assumption F ∩ Θ null = ∅ implies that q ∈ supp ( η − C ) , a contradiction. Similarly,we have that G ∩ Θ null = ∅ because [ C ] ∈ M g − can be assumed to have no eventheta-characteristics η + C ∈ Pic g − ( C ) with h ( C, η + C ) ≥ , that is [ C, η + C ] / ∈ Θ null ⊂ S + g − .Finally, we assume that there exists a point [ X := C ∪ { y,q } E, η C = η + C , η E = O E (1)] ∈ H ∩ Θ null . Then certainly h ( X, η X ) ≥ and from the Mayer-Vietoris sequence on X we find that H ( X, η X ) = Ker { H ( C, η C ) ⊕ H ( E, O E (1)) → C y,q } , hence h ( C, η C ) = h ( X, η X ) ≥ . This contradicts the assumption that [ C ] ∈ M g − is general. A similar argument works for the special point in H ∩ π − (∆ ) , hence H · Θ null = 0 . (cid:3) Proof of Theorem 0.2 . Looking at the expansion of [Θ null ] , Theorem 2.3 gives the relations F · Θ null = ¯ λ −
12 ¯ α + ¯ β = 0 , G · Θ null = 3¯ λ −
12 ¯ α −
12 ¯ β + 3 ¯ α = 0 and H · Θ null = ( g −
1) ¯ β − ¯ α = 0 . Since we have already computed ¯ α i = 0 and ¯ β i = 1 / for ≤ i ≤ [ g/ , (cf. Theorem2.2), we obtain that ¯ λ = 1 / , ¯ α = 1 / and ¯ β = 0 . This completes the proof. (cid:3) A consequence of Theorem 0.2 is a new proof of the main result from [T]: heorem 2.4. If M g is the locus of curves [ C ] ∈ M g with a vanishing theta-null then itsclosure has class equal to M g ≡ g − (cid:16) (2 g + 1) λ − g − δ − [ g/ X i =1 (2 g − i − i − δ i (cid:17) ∈ Pic( M g ) . Proof.
We use the scheme-theoretic equality π ∗ (Θ null ) = M g as well as the formulas π ∗ ( λ ) = 2 g − (2 g + 1) λ, π ∗ ( α ) = 2 g − δ , π ∗ ( β ) = 2 g − (2 g − + 1) δ , π ∗ ( α i ) = 2 g − (2 i +1)(2 g − i + 1) δ i and π ∗ ( β i ) = 2 g − (2 i − g − i − δ i valid for ≤ i ≤ [ g/ . (cid:3) R EFERENCES [AJ] D. Abramovich and T. Jarvis,
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