THE BIRATIONAL TYPE OF THE MODULI SPACE OF EVEN SPIN CURVES
GAVRIL FARKAS
The moduli space Sg of smooth spin curves parameterizes pairs [C, η], where [C] ∈ Mg is a curve of genusgandη ∈ Picg−1(C) is a theta-characteristic. The finite forgetful mapπ :Sg → Mg has degree22gandSgis a disjoint union of two connected componentsSg+andSg−of relative degrees2g−1(2g+ 1)and2g−1(2g−1)corresponding to even and odd theta-characteristics respectively. A compactificationSgofSgoverMg
is obtained by considering the coarse moduli space of the stack of stable spin curves of genusg(cf. [C], [CCC] and [AJ]). The projectionSg → Mgextends to a finite branched coveringπ :Sg → Mg. In this paper we determine the Kodaira dimension ofS+g: Theorem 0.1. The moduli spaceS+g of even spin curves is a variety of general type forg >8 and it is uniruled forg <8. The Kodaira dimension ofS+8 is non-negative1.
It was classically known that S+2 is rational. The Scorza map establishes a bira- tional isomorphism betweenS+3 andM3, cf. [DK], henceS+3 is rational. Very recently, Takagi and Zucconi [TZ] showed thatS+4 is rational as well. Theorem 0.1 can be com- pared to [FL] Theorem 0.3: The moduli spaceRgof Prym varieties of dimensiong−1 (that is, non-trivial square roots ofOCfor each[C]∈ Mg) is of general type wheng >13 andg 6= 15. On the other handRg is unirational forg < 8. Surprisingly, the problem of determining the Kodaira dimension has a much shorter solution forS+g than forRg
and our results are complete.
We describe the strategy to prove that S+g is of general type for a given g. We denote byλ=π∗(λ)∈Pic(S+g)the pull-back of the Hodge class and byα0, β0 ∈Pic(S+g) andαi, βi∈Pic(S+g)for1≤i≤[g/2]boundary divisor classes such that
π∗(δ0) =α0+ 2β0and π∗(δi) =αi+βi for1≤i≤[g/2]
(see Section 2 for precise definitions). Using Riemann-Hurwitz and [HM] we find that KS+
g ≡π∗(KMg) +β0 ≡13λ−2α0−3β0−2
[g/2]X
i=1
(αi+βi)−(α1+β1).
We prove that K
S+g is a big Q-divisor class by comparing it against the class of the closure inS+g of the divisorΘnullonSg+of non-vanishing even theta characteristics:
Research partially supported by an Alfred P. Sloan Fellowship.
1Building 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.
Theorem 0.2. The closure in S+g of the divisorΘnull := {[C, η] ∈ Sg+ : H0(C, η) 6= 0} of non-vanishing even theta characteristics has class equal to
Θnull≡ 1 4λ− 1
16α0− 1 2
[g/2]X
i=1
βi ∈ Pic(S+g).
Note that the coefficients ofβ0andαi for1≤i≤[g/2]in the expansion of[Θnull] are equal to 0. To prove Theorem 0.2, one can use test curves onS+g or alternatively, realize Θnull as the push-forward of the degeneracy locus of a map of vector bundles of the same rank defined over a certain Hurwitz scheme coveringS+g and use [F1] and [F2] to compute the class of this locus. Then we use [FP] Theorem 1.1, to construct for each genus3 ≤ g ≤ 22an effective divisor classD ≡ aλ−P[g/2]
i=0 biδi ∈ Eff(Mg)with coefficients satisfying the inequalities
a b0 ≤
6 +g+112 , ifg+ 1is composite 7, ifg= 10
6k2+k−6
k(k−1) , ifg= 2k−2≥4
andbi/b0 ≥4/3for1≤i≤[g/2]. Wheng+ 1is composite we choose forDthe closure of the Brill-Noether divisor of curves with agrd, that is,Mrg,d:={[C]∈ Mg:Grd(C)6=∅}
in case when the Brill-Noether numberρ(g, r, d) =−1, and then cf. [EH2]
Mrg,d≡cg,d,r
(g+ 3)λ−g+ 1 6 δ0−
[g/2]X
i=1
i(g−i)δi
∈Pic(Mg).
Forg= 10we take the closure of the divisorK10:={[C]∈ M10:Clies on aK3surface}
(cf. [FP] Theorem 1.6). In the remaining cases, when necessarilyg= 2k−2, we choose forD the Gieseker-Petri divisorGP1g,k consisting of those curves[C] ∈ Mg such that there exists a pencilA∈Wk1(C)such that the multiplication map
µ0(A) :H0(C, A)⊗H0(C, KC ⊗A∨)→H0(C, KC)
is not an isomorphism, see [EH2], [F2]. Having chosenD, we form theQ-linear combi- nation of divisor classes
8·Θnull+ 3
2b0·π∗(D) = 2 + 3a 2b0
λ−2α0−3β0−
[g/2]X
i=1
3bi 2b0αi−
[g/2]X
i=1
4 +3bi 2
βi ∈Pic(S+g),
from which we can write
KS+g =νg·λ+ 8Θnull+ 3 2b0
π∗(D) +
[g/2]X
i=1
ci·αi+c′i·βi),
whereci, c′i ≥ 0. Moreoverνg >0precisely wheng ≥ 9, whileν8 = 0. Since the class λ ∈ Pic(S+g)is big and nef, we obtain thatKS+
g is a bigQ-divisor class on the normal varietyS+g as soon asg > 8. It is proved in [Lud] that for g ≥ 4pluricanonical forms defined onS+g,regextend to any resolution of singularitiesScg+→ S+g, which shows that S+g is of general type wheneverνg >0and completes the proof of Theorem 0.1 forg≥8.
When g ≤ 7 we show that KS+
g ∈/ Eff(S+g) by constructing a covering curve R ⊂ S+g
such that R·K
S+g < 0, cf. Theorem 1.2. We then use [BDPP] to conclude thatS+g is uniruled.
I would like to thank the referee for pertinent comments which led to a clearly improved version of this paper.
1. THE STACK OF SPIN CURVES
We review a few facts about Cornalba’s compactification π :Sg → Mg, see [C].
IfX is a nodal curve, a smooth rational componentE ⊂ X is said to beexceptional if
#(E∩X−E) = 2. The curveXis said to bequasi-stableif#(E∩X−E) ≥2for any smooth rational componentE ⊂X, and moreover any two exceptional components of Xare disjoint. A quasi-stable curve is obtained from a stable curve by blowing-up each node at most once. We denote by[st(X)]∈ Mgthe stable model ofX.
Definition 1.1. Aspin curveof genusgconsists of a triple(X, η, β), whereXis a genusg quasi-stable curve,η∈Picg−1(X)is a line bundle of degreeg−1such thatηE =OE(1) for every exceptional componentE ⊂X, andβ :η⊗2 → ωX is a sheaf homomorphism which is generically non-zero along each non-exceptional component ofX.
Afamily 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⊗2 →ωX is a sheaf homomorphism, such that for every points∈Sthe restriction (Xs, ηXs, βXs :ηX⊗2s →ωXs)is a spin curve.
To describe locally the map π : Sg → Mg we follow [C] Section 5. We fix [X, η, β]∈ Sg and setC :=st(X). We denote byE1, . . . , Erthe exceptional components ofXand byp1, . . . , pr ∈ Csing the nodes which are images of exceptional components.
The automorphism group of(X, η, β)fits in the exact sequence of groups 1−→Aut0(X, η, β) −→Aut(X, η, β)−→resC Aut(C).
We denote byC3g−3τ the versal deformation space of(X, η, β) where for1 ≤i ≤ r the locus(τi = 0) ⊂ C3g−3τ corresponds to spin curves in which the componentEi ⊂ X persists. Similarly, we denote byC3g−3t = Ext1(ΩC,OC) the versal deformation space ofCand denote by(ti = 0)⊂C3g−3t the locus where the nodepi ∈Cis not smoothed.
Then around the point[X, η, β], the morphismπ:Sg→ Mgis locally given by the map (1) C3g−3τ
Aut(X, η, β) → C3g−3t
Aut(C), ti=τi2 (1≤i≤r) and ti=τi (r+ 1≤i≤3g−3).
From now on we specialize to the case of even spin curves and describe the boundary ofS+g. In the process we determine the ramification of the finite coveringπ:S+g → Mg. 1.1. The boundary divisors ofS+g.
If[X, η, β] ∈π−1([C∪yD])where[C, y]∈ Mi,1 and[D, y]∈ Mg−i,1, then neces- sarilyX:=C∪y1E∪y2D, whereEis an exceptional component such thatC∩E ={y1} andD∩E ={y2}. Moreover
η= ηC, ηD, ηE =OE(1)
∈Picg−1(X),
whereηC⊗2 =KC, η⊗2D =KD. The conditionh0(X, η) ≡0mod2, implies that the theta- characteristics ηC andηD have the same parity. We denote byAi ⊂ S+g the closure of the locus corresponding to pairs([C, y, ηC],[D, y, ηD]) ∈ Si,1+ × Sg−i,1+ and byBi ⊂ S+g
the closure of the locus corresponding to pairs([C, y, ηC],[D, y, ηD])∈ Si,1− × Sg−i,1− . For a general point[X, η, β]∈Ai∪Biwe have that Aut0(X, η, β) =Aut(X, η, β) = Z2. Using (1), the map C3g−3τ → C3g−3t is given by t1 = τ12 and ti = τi for i ≥ 2. Furthermore, Aut0(X, η, β) acts onC3g−3τ via (τ1, τ2, . . . , τ3g−3) 7→ (−τ1, τ2, . . . , τ3g−3). It follows that∆i ⊂ Mg is not a branch divisor forπ : S+g → Mg and ifαi = [Ai] ∈ Pic(S+g)andβi= [Bi]∈Pic(S+g), then for1≤i≤[g/2]we have the relation
(2) π∗(δi) =αi+βi.
Moreover,π∗(αi) = 2g−2(2i+ 1)(2g−i+ 1)δiandπ∗(βi) = 2g−2(2i−1)(2g−i−1)δi. For a point[X, η, β]such thatst(X) = Cyq := C/y ∼ q, with[C, y, q] ∈ Mg−1,2, there are two possibilities depending on whetherX possesses an exceptional compo- nent or not. IfX =Cyq andηC := ν∗(η)whereν :C → X denotes the normalization map, thenηC⊗2 =KC(y+q). For each choice ofηC ∈ Picg−1(C)as above, there is pre- cisely one choice of gluing the fibresηC(y)andηC(q)such thath0(X, η)≡0mod2. We denote byA0 the closure inS+g of the locus of points[Cyq, ηC ∈p
KC(y+q)]as above and clearly deg(A0/∆0) = 22g−2.
IfX =C∪{y,q}E whereE is an exceptional component, thenηC := η⊗ OC is a theta-characteristic onC. SinceH0(X, ω) ∼= H0(C, ωC), it follows that[C, ηC] ∈ Sg−1+ . For[C, y, q]∈ Mg−1,2sufficiently generic we have that Aut(X, η, β) =Aut(C) ={IdC}, and then from (1) it follows thatπ is simply branched over such points. We denote by B0 ⊂ S+g the closure of the locus of points [C ∪{y,q}E, ηC ∈ √
KC, ηE = OE(1)]. If α0 = [A0]∈Pic(S+g)andβ0 = [B0]∈Pic(S+g), we then have the relation
(3) π∗(δ0) =α0+ 2β0.
Note thatπ∗(α0) = 22g−2δ0andπ∗(β0) = 2g−2(2g−1+ 1)δ0. 1.2. The uniruledness ofS+g for smallg.
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 thatκ(Rg) =−∞forg <8, cf. [FL] Theorem 0.7.
Theorem 1.2. Forg <8, the spaceS+g is uniruled.
Proof. We start with a fixedK3surfaceScarrying a Lefschetz pencil of curves of genus g. This induces a fibration f : Blg2(S) → P1 and then we setB := mf
∗(P1) ⊂ Mg, where mf : P1 → Mg is the moduli map mf(t) := [f−1(t)]. We have the following well-known formulas onMg(cf. [FP] Lemma 2.4):
B·λ=g+ 1, B·δ0 = 6g+ 18, andB·δi = 0 fori≥1.
We liftB to a pencilR⊂ S+g of spin curves by taking
R:=B×Mg S+g ={[Ct, ηCt]∈ S+g : [Ct]∈B, ηCt ∈Picg−1(Ct), t∈P1} ⊂ S+g.
Using (3) one computes the intersection numbers with the generators of Pic(S+g):
R·λ= (g+ 1)2g−1(2g+ 1), R·α0 = (6g+ 18)22g−2 andR·β0 = (6g+ 18)2g−2(2g−1+ 1).
Furthermore,Ris disjoint from all the remaining boundary classes ofS+g, that is,R·αi = R·βi = 0for1≤i≤[g/2]. One verifies thatR·K
S+g <0precisely wheng≤7. SinceR is a covering curve forS+g in the rangeg≤7, we find thatKS+
g is not pseudo-effective, that is, K
S+g ∈ Eff(S+g)c. Pseudo-effectiveness of the canonical bundle is a birational property 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 thatS+g is uniruled forg≤7.
2. THE GEOMETRY OF THE DIVISOR Θnull
We compute the class of the divisorΘnullusing test curves. The same calculation can be carried out using techniques developed in [F1], [F2] to calculate push-forwards of tautological classes from stacks of limit linear seriesgrd(see also Remark 2.1).
Forg ≥9, Harer [H] has showed thatH2(Sg+,Q) ∼=Q. The range for which this result holds has been recently improved tog ≥ 5 in [P]. In particular, it follows that Pic(S+g)Q is generated by the classesλ, αi, βi fori = 0, . . . ,[g/2]. Thus we can expand the divisor classΘnullin terms of the generators of the Picard group
(4) Θnull≡¯λ·λ−α¯0·α0−β¯0·β0−
[g/2]
X
i=1
¯
αi·αi+ ¯βi·βi
∈Pic(S+g)Q, and determine the coefficients¯λ,α¯0,β¯0,α¯i andβ¯i ∈Qfor1≤i≤[g/2].
Remark 2.1. To show that the class [Θnull] ∈ Pic(Sg+)Q is a multiple ofλand thus, the expansion (4) makes sense for allg ≥ 3, one does not need to know that Pic(Sg+)Q is infinite cyclic. For instance, for eveng = 2k−2 ≥ 4, we note that, via the base point free pencil trick,[C, η]∈Θnullif and only if the multiplication map
µC(A, η) :H0(C, A)⊗H0(C, A⊗η)→H0(C, A⊗2⊗η)
is not an isomorphism for a base point free pencil A ∈ Wk1(C). We setMfg to be the open subvariety consisting of curves[C] ∈ Mg such thatWk−11 (C) = ∅and denote by σ:G1k →Mfgthe Hurwitz scheme of pencilsg1kand by
τ :G1k×Mfg Sg+→ Sg+, u:G1k×Mfg Sg+→G1k the (generically finite) projections. ThenΘnull=τ∗(Z), where
Z ={[A, C, η]∈G1k×Mfg Sg+:µC(A, η)is not injective}.
Via this determinantal presentation, the class of the divisorZis expressible as a combi- nation ofτ∗(λ), u∗(a), u∗(b), wherea,b ∈ Pic(G1k)Q are the tautological classes defined in e.g. [FL] p.15. Sinceτ∗(u∗(a)) =π∗(σ∗(a))(and similarly for the classb), the conclu- sion follows. For odd genusg= 2k−1, one uses a similar argument replacingG1kwith any generically finite covering ofMggiven by a Hurwitz scheme (for instance, we take the space of pencilsg1k+1with a triple ramification point).
We start the proof of Theorem 0.2 by determining the coefficients of αi and βi
(i≥1)in the expansion of[Θnull].
Theorem 2.2. We fix integersg≥3and1≤i≤[g/2]. The coefficient ofαi in the expansion of[Θnull]equals0, while the coefficient ofβi equals−1/2. That is,α¯i = 0andβ¯i = 1/2.
Proof. For each integer2≤i≤g−1, we fix general curves[C]∈ Miand[D, q]∈ Mg−i,1
and consider the test curve Ci := {C ∪y∼q D}y∈C ⊂ ∆i ⊂ Mg. We lift Ci to test curvesFi ⊂Ai andGi ⊂Bi insideS+g constructed as follows. We fix even (resp. odd) theta-characteristicsηC+ ∈ Pici−1(C)andη+D ∈ Picg−i−1(D)(resp. ηC− ∈ Pici−1(C) and ηD−∈Picg−i−1(D)).
If E ∼= P1 is an exceptional component, we define the family Fi (resp. Gi) as consisting of spin curves
Fi:=
t:= [C∪yE∪qD, ηC =ηC+, ηE =OE(1), ηD =η+D]∈ S+g :y∈C and
Gi:=
t:= [C∪yE∪qD, ηC =ηC−, ηE =OE(1), ηD =η−D]∈ S+g :y∈C . Sinceπ∗(Fi) = π∗(Gi) = Ci, clearly Fi·αi = Ci·δi = 2−2i, Fi ·βi = 0 andFi has intersection number0with all other generators of Pic(S+g). Similarly
Gi·βi = 2−2i, Gi·αi= 0, Gi·λ= 0, andGidoes not intersect the remaining boundary classes inS+g.
Next we determineFi∩Θnull. Assume that a pointt∈Fi lies inΘnull. Then there exists a family of even spin curves(f :X →S, η, β), whereS =Spec(R), withRbeing a discrete valuation ring andX is a smooth surface, such that, if0, ξ ∈ S denote the special and the generic point ofSrespectively andXξis the generic fibre off, then
h0(Xξ, ηξ)≥2, h0(Xξ, ηξ)≡0mod 2, ηξ⊗2∼=ωXξ and f−1(0), ηf−1(0)
=t∈ S+g. Following the procedure described in [EH1] p. 347-351, this data produces a limit linear seriesg1g−1 onC∪D, say
l:=
lC = (LC, VC), lD = (LD, VD)
∈G1g−1(C)×G1g−1(D),
such that the underlying line bundlesLC andLD respectively, are obtained from the line bundle(ηC+, ηE, ηD+)by dropping theE-aspect and then tensoring the line bundles ηC+andη+D by line bundles supported at the pointsy ∈ C andq ∈ Drespectively. For degree reasons, it follows thatLC =ηC+⊗ OC((g−i)y) andLD = ηD+⊗ OD(iq). Since bothCandDare general in their respective moduli spaces, we have thatH0(C, η+C) = 0 and H0(D, ηD+) = 0. In particular al1C(y) ≤ g −i−1 and al0D(q) < al1D(q) ≤ i−1, hence al1C(y) +al0D(q) ≤ g−2, which contradicts the definition of a limit g1g−1. Thus Fi∩Θnull=∅. This implies thatα¯i = 0, for all1≤i≤[g/2](fori= 1, one uses instead the curveFg−1 ⊂A1 to reach the same conclusion).
Assume that t ∈ Gi∩Θnull. By the same argument as above, retaining also the notation, there is an induced limit linear series onC∪D,
(lC, lD)∈G1g−1(C)×G1g−1(D),
whereLC = ηC−⊗ OC((g−i)y) andLD =η−D⊗ OD(iq). Since[C]∈ Mi and[D, q]∈ Mg−i,1are both general, we may assume thath0(D, η−D) =h0(C, η−C) = 1,q /∈supp(ηD−) and that supp(η−C) consists of i−1 distinct points. In particular al1D(q) ≤ i, hence al0C(y) ≥g−1−al1D(q) ≥g−i−1. Sinceh0(C, ηC−) = 1, it follows that one has in fact equality, that is,al0C(y) =g−i−1and then necessarilyal1D(q) =i.
Similarly,al1C(y)≤g−i+ 1(otherwise div(η−C)≥2y, that is, supp(ηC−)would be non-reduced, a contradiction), thusal0D(q)≥i−2, and the last two inequalities must be equalities as well (one uses thath0 D, LD ⊗ OD(−(i−1)q)
=h0(D, ηD−⊗ OD(q)) = 1, that is,al0D(q)< i−1). Sinceal1C(y) =g−i+ 1, we find thaty∈supp(η−C).
To sum up, we have showed that(lC, lD)is a refined limitg1g−1and in fact (5) lD =|ηD−⊗OD(2q)|+(i−2)·q ∈G1g−1(D), lC =|η−C⊗OC(y)|+(g−i−1)·y ∈G1g−1(C), hencealD(q) = (i−2, i)andalC(y) = (g−i−1, g−i+ 1).
To prove that the intersection between Gi and Θnull is transversal, we follow closely [EH3] Lemma 3.4 (see especially theRemarkon p. 45): The restrictionΘnull|Gi is isomorphic, as a scheme, to the varietyτ :T1g−1(Gi) →Gi of limit linear seriesg1g−1on the curves of compact type{C∪y∼qD:y∈C}, whoseCandD-aspects are obtained by twisting suitably aty ∈ Candq ∈ Dthe fixed theta-characteristicsηC−andη−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 becauseGi consists entirely of singular spin curves of compact type, the scheme T1g−1(Gi) splits as a product of the corresponding moduli spaces of C andD-aspects respectively of the limits g1g−1. By direct calculation we have showed thatT1g−1(Gi) ∼= supp(η−C)× {lD}. Since supp(η−C) is a reduced0-dimensional scheme, we obtain thatΘnull|Gi is everywhere reduced. It follows thatGi·Θnull = #supp(ηC−) = i−1and thenβ¯i = (Gi·Θnull)/(2i−2). This argument does not work fori= 1, when one uses instead the intersection ofΘnullwith
Gg−1, and this finishes the proof.
Next we construct two pencils in S+g which are lifts of the standard degree12 pencil of elliptic tails inMg. We fix a general pointed curve[C, q]∈ Mg−1,1and a pencil f : Bl9(P2) → P1 of plane cubics together with a sectionσ :P1 → Bl9(P2)induced by one of the base points. We then consider the pencilR:={[C∪q∼σ(λ)f−1(λ)]}λ∈P1 ⊂ Mg. We fix an odd theta-characteristic ηC− ∈ Picg−2(C) such thatq /∈ supp(ηC−) and E ∼=P1will again denote an exceptional component. We define the family
F0 :={[C∪qE∪σ(λ)f−1(λ), ηC =ηC−, ηE =OE(1), ηf−1(λ) =Of−1(λ)] :λ∈P1} ⊂ S+g. SinceF0∩A1 =∅, we find thatF0·β1=π∗(F0)·δ1 =−1. Similarly,F0·λ=π∗(F0)·λ= 1 and obviouslyF0·αi =F0·βi = 0for2 ≤i≤[g/2]. For each of the12pointsλ∞∈P1 corresponding to singular fibres ofR, the associatedηλ∞ ∈ Picg−1(C ∪E∪f−1(λ∞)) are actual line bundles onC∪E∪f−1(λ∞)(that is, we do not have to blow-up the extra node). Thus we obtain thatF0·β0 = 0, thereforeF0·α0 =π∗(F0)·δ0 = 12.
We also fix an even theta-characteristicηC+∈Picg−2(C)and consider the degree3 branched coveringγ :S+1,1 → M1,1forgetting the spin structure. We define the pencil G0:={
C∪qE∪σ(λ)f−1(λ), ηC =η+C, ηE =OE(1), ηf−1(λ) ∈γ−1[f−1(λ)]
:λ∈P1} ⊂ S+g. Sinceπ∗(G0) = 3R, we have that G0 ·λ = 3. Obviously G0 ·β0 = G0·β1 = 0, hence G0 ·α1 = π∗(G0)·δ1 = −3. The map γ : S+1,1 → M1,1 is simply ramified over the point corresponding toj-invariant∞. Hence,G0 ·α0 = 12andG0 ·β0 = 12, which is consistent with formula (3).
The last pencil we construct lies in the boundary divisorB0⊂ S+g: SettingE ∼=P1 for an exceptional component, we define
H0 :={[C∪{y,q}E, ηC =ηC+, ηE =OE(1)] :y∈C} ⊂ S+g. The fibre ofH0over the pointy=q∈Cis the even spin curve
C∪qE′∪q′ E′′∪{q′′,y′′}E, ηC =η+C, ηE′ =OE′(1), ηE =OE(1), ηE′′=OE′′(−1) , having as stable model[C∪qE∞], whereE∞:=E′′/y′′ ∼q′′is the rational nodal curve corresponding toj = ∞. HereE′, E′′ are rational curves,E′ ∩E′′ = {q′}, E ∩E′′ = {q′′, y′′}and the stabilization map forC∪E∪E′∪E′′contracts the componentsE′and E, while identifyingq′′andy′′.
We find thatH0·λ= 0, H0·αi =H0·βi= 0for2≤i≤[g/2]. MoreoverH0·α0 = 0, henceH0·β0 = 12π∗(H0)·δ0= 1−g. Finally,H0·α1= 1andH0·β1= 0.
Theorem 2.3. IfF0, G0, H0 ⊂ S+g are the families of spin curves defined above, then F0·Θnull=G0·Θnull=H0·Θnull= 0.
Proof. From the limit linear series argument in the proof of Theorem 2.2 we get that the assumptionF0 ∩Θnull 6= ∅ implies that q ∈ supp(η−C), a contradiction. Similarly, we have that G0 ∩Θnull = ∅ because [C] ∈ Mg−1 can be assumed to have no even theta-characteristicsη+C ∈Picg−2(C)withh0(C, ηC+) ≥2, that is [C, ηC+]∈/ Θnull ⊂ S+g−1. Finally, we assume that there exists a point[X := C∪{y,q}E, ηC = ηC+, ηE = OE(1)] ∈ H0∩Θnull. Then certainly h0(X, ηX) ≥ 2and from the Mayer-Vietoris sequence onX we find that
H0(X, ηX) =Ker{H0(C, ηC)⊕H0(E,OE(1))→C2y,q},
hence h0(C, ηC) = h0(X, ηX) ≥ 2. This contradicts the assumption that[C] ∈ Mg−1
is general. A similar argument works for the special point in H0 ∩π−1(∆1), hence
H0·Θnull= 0.
Proof of Theorem 0.2. Looking at the expansion of[Θnull], Theorem 2.3 gives the relations F0·Θnull= ¯λ−12¯α0+ ¯β1 = 0, G0·Θnull= 3¯λ−12¯α0−12 ¯β0+ 3¯α1 = 0
and H0·Θnull= (g−1) ¯β0−α¯1= 0.
Since we have already computedα¯i = 0andβ¯i = 1/2for1 ≤ i≤ [g/2], (cf. Theorem 2.2), we obtain thatλ¯= 1/4,α¯0 = 1/16andβ¯0 = 0. This completes the proof.
A consequence of Theorem 0.2 is a new proof of the main result from [T]:
Theorem 2.4. If M1g is the locus of curves [C] ∈ Mg with a vanishing theta-null then its closure has class equal to
M1g ≡2g−3
(2g+ 1)λ−2g−3δ0−
[g/2]X
i=1
(2g−i−1)(2i−1)δi
∈Pic(Mg).
Proof. We use the scheme-theoretic equality π∗(Θnull) = M1g as well as the formulas π∗(λ) = 2g−1(2g+ 1)λ, π∗(α0) = 22g−2δ0, π∗(β0) = 2g−2(2g−1+ 1)δ0, π∗(αi) = 2g−2(2i+ 1)(2g−i+ 1)δi andπ∗(βi) = 2g−2(2i−1)(2g−i−1)δivalid for1≤i≤[g/2].
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HUMBOLDT-UNIVERSITAT ZU¨ BERLIN, INSTITUTF ¨URMATHEMATIK, 10099 BERLIN
E-mail address:farkas@math.hu-berlin.de
