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−1}(C) is a theta-characteristic. The finite forgetful map π : S_{g} → M_{g} has degree 2^{2g} and S_{g} is a disjoint union of two connected components S_{g}^{+} and S_{g}^{−} of relative degrees 2^{g−1}(2^{g} + 1) and 2^{g−1}(2^{g} − 1) corresponding to even and odd theta-characteristics respectively. A compactification \overline{S}_{g} of S_{g} over \overline{M}_{g} is obtained by considering the coarse moduli space of the stack of stable spin curves of genus g (cf. [C], [CCC] and [AJ]). The projection S_{g} → M_{g} extends to a finite branched covering π : \overline{S}_{g} → \overline{M}_{g}. In this paper we determine the Kodaira dimension of \overline{S}_{g}^{+}:

Theorem 0.1. The moduli space \overline{S}_{g}^{+} of even spin curves is a variety of general type for g > 8 and it is uniruled for g < 8. The Kodaira dimension of \overline{S}_{8}^{+} is non-negative^{1}.

It was classically known that \overline{S}_{2}^{+} is rational. The Scorza map establishes a birational isomorphism between \overline{S}_{3}^{+} and \overline{M}_{3}, cf. [DK], hence \overline{S}_{3}^{+} is rational. Very recently, Takagi and Zucconi [TZ] showed that \overline{S}_{4}^{+} is rational as well. Theorem 0.1 can be compared to [FL] Theorem 0.3: The moduli space \overline{R}_{g} of Prym varieties of dimension g − 1 (that is, non-trivial square roots of O_{C} for each [C] ∈ M_{g}) is of general type when g > 13 and g ≠ 15. On the other hand \overline{R}_{g} is unirational for g < 8. Surprisingly, the problem of determining the Kodaira dimension has a much shorter solution for \overline{S}_{g}^{+} than for \overline{R}_{g} and our results are complete.

We describe the strategy to prove that \overline{S}_{g}^{+} is of general type for a given g. We denote by λ = π^{∗}(λ) ∈ Pic(\overline{S}_{g}^{+}) the pull-back of the Hodge class and by α_{0}, β_{0} ∈ Pic(\overline{S}_{g}^{+}) and α_{i}, β_{i} ∈ Pic(\overline{S}_{g}^{+}) for 1 ≤ i ≤ [g/2] boundary divisor classes such that

π^{∗}(δ_{0}) = α_{0} + 2β_{0} and π^{∗}(δ_{i}) = α_{i} + β_{i} for 1 ≤ i ≤ [g/2]

(see Section 2 for precise definitions). Using Riemann-Hurwitz and [HM] we find that

K\overline{S}{+ \atop g} ≡ π^{∗}(K_{\overline{M}_{g}}) + β_{0} ≡ 13λ − 2α_{0} − 3β_{0} − 2 ∑_{i=1}^{[g/2]}(α_{i} + β_{i}) − (α_{1} + β_{1}).

We prove that K\overline{S}{+ \atop g} is a big ℚ-divisor class by comparing it against the class of the

closure in \overline{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.

1Building on the results of this paper, we have proved quite recently in joint work with A. Verra, that κ(\overline{S}_{8}^{+}) = 0. Details will appear later.

Theorem 0.2. The closure in \overline{S}_{g}^{+} of the divisor Θ_{null} := \{[C, η] ∈ S_{g}^{+} : H^{0}(C, η) ≠ 0\} of non-vanishing even theta characteristics has class equal to

\overline{Θ}_{null} ≡ \frac{1}{4}λ − \frac{1}{16}α_{0} − \frac{1}{2} ∑_{i=1}^{[g/2]} β_{i} ∈ Pic(\overline{S}_{g}^{+}).

Note that the coefficients of β_{0} and α_{i} for 1 ≤ i ≤ [g/2] in the expansion of [\overline{Θ}_{null}] are equal to 0. To prove Theorem 0.2, one can use test curves on \overline{S}_{g}^{+} or alternatively, realize \overline{Θ}_{null} as the push-forward of the degeneracy locus of a map of vector bundles of the same rank defined over a certain Hurwitz scheme covering \overline{S}_{g}^{+} and use [F1] and

[F2] to compute the class of this locus. Then we use [FP] Theorem 1.1, to construct for each genus 3 ≤ g ≤ 22 an effective divisor class D ≡ aλ − ∑_{i=0}^{[g/2]} b_{i}δ_{i} ∈ Eff(\overline{M}_{g}) with coefficients satisfying the inequalities

\frac{a}{b_{0}} ≤ \begin{cases}6 + \frac{12}{g+1}, & if g + 1 is composite \\ 7, & if g = 10 \\ \frac{6k^{2}+k−6}{k(k−1)}, & if g = 2k − 2 ≥ 4 \end{cases}

and b_{i}/b_{0} ≥ 4/3 for 1 ≤ i ≤ [g/2]. When g + 1 is composite we choose for D the closure of the Brill-Noether divisor of curves with a 𝔤_{d}^{r}, that is, M_{g,d}^{r} := {[C] ∈ M_{g} : G_{d}^{r}(C) ≠ ∅} in case when the Brill-Noether number ρ(g, r, d) = −1, and then cf. [EH2]

\overline{M}_{g,d}^{r} ≡ c_{g,d,r}((g + 3)λ − \frac{g + 1}{6}δ_{0} − ∑_{i=1}^{[g/2]} i(g − i)δ_{i}) ∈ Pic(\overline{M}_{g}).

For g = 10 we take the closure of the divisor K_{10} := {[C] ∈ M_{10} : C lies on a K3 surface} (cf. [FP] Theorem 1.6). In the remaining cases, when necessarily g = 2k − 2, we choose for D the Gieseker-Petri divisor \overline{GP}_{g,k}^{1} consisting of those curves [C] ∈ M_{g} such that there exists a pencil A ∈ W_{k}^{1}(C) such that the multiplication map

μ_{0}(A) : H^{0}(C, A) ⊗ H^{0}(C, K_{C} ⊗ A^{∨}) → H^{0}(C, K_{C})

is not an isomorphism, see [EH2], [F2]. Having chosen D, we form the ℚ-linear combination of divisor classes

8 · \overline{Θ}_{null} + \frac{3}{2b_{0}} · π^{∗}(D) = (2 + \frac{3a}{2b_{0}} )λ − 2α_{0} − 3β_{0} − ∑_{i=1}^{[g/2]} \frac{3b_{i}}{2b_{0}} α_{i} − ∑_{i=1}^{[g/2]}(4 + \frac{3b_{i}}{2} )β_{i} ∈ Pic(\overline{S}_{g}^{+}),

from which we can write

K\overline{S}{+ \atop g} = ν_{g} · λ + 8\overline{Θ}_{null} + \frac{3}{2b_{0}} π^{∗}(D) + ∑_{i=1}^{[g/2]}(c_{i} · α_{i} + c′_{i} · β_{i}),

where c_{i}, c′_{i} ≥ 0. Moreover ν_{g} > 0 precisely when g ≥ 9, while ν_{8} = 0. Since the class λ ∈ Pic(\overline{S}_{g}^{+}) is big and nef, we obtain that K\overline{S}{+ \atop g} is a big ℚ-divisor class on the normal

variety \overline{S}_{g}^{+} as soon as g > 8. It is proved in [Lud] that for g ≥ 4 pluricanonical forms

defined on \overline{S}_{g,reg}^{+} extend to any resolution of singularities Ŝ_{g}^{+} → \overline{S}_{g}^{+}, which shows that \overline{S}_{g}^{+} is of general type whenever ν_{g} > 0 and completes the proof of Theorem 0.1 for g ≥ 8.

When g ≤ 7 we show that K\overline{S}{+ \atop g} ∉ \overline{Eff}(\overline{S}_{g}^{+}) by constructing a covering curve R ⊂ \overline{S}_{g}^{+}

such that R · K\overline{S}{+ \atop g} < 0, cf. Theorem 1.2. We then use [BDPP] to conclude that \overline{S}_{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 : \overline{S}_{g} → \overline{M}_{g}, see [C]. If X is a nodal curve, a smooth rational component E ⊂ X is said to be exceptional if #(E ∩ \overline{X − E}) = 2. The curve X is said to be quasi-stable if #(E ∩ \overline{X − E}) ≥ 2 for any smooth 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 each node at most once. We denote by [st(X)] ∈ \overline{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−1}(X) is a line bundle of degree g − 1 such that η_{E} = O_{E}(1) for every exceptional component E ⊂ X, and β : η^{⊗2} → ω_{X} is a sheaf homomorphism which 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^{⊗2} → 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}}^{⊗2} → ω_{X_{s}}) is a spin curve.

To describe locally the map π : \overline{S}_{g} → \overline{M}_{g} we follow [C] Section 5. We fix [X, η, β] ∈ \overline{S}_{g} and set C := st(X). We denote by E_{1}, …, E_{r} the exceptional components of X and by p_{1}, …, 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

1 ⟶ Aut_{0}(X, η, β) ⟶ Aut(X, η, β) ⟶^{res_{C}} Aut(C).

We denote by ℂ_{τ}^{3g−3} the versal deformation space of (X, η, β) where for 1 ≤ i ≤ r the locus (τ_{i} = 0) ⊂ ℂ_{τ}^{3g−3} corresponds to spin curves in which the component E_{i} ⊂ X persists. Similarly, we denote by ℂ_{t}^{3g−3} = Ext^{1}(Ω_{C}, O_{C}) the versal deformation space of C and denote by (t_{i} = 0) ⊂ ℂ_{t}^{3g−3} the locus where the node p_{i} ∈ C is not smoothed. Then around the point [X, η, β], the morphism π : \overline{S}_{g} → \overline{M}_{g} is locally given by the map

(1) \frac{ℂ_{τ}^{3g−3}}{Aut(X, η, β)} → \frac{ℂ_{t}^{3g−3}}{Aut(C)}, t_{i} = τ_{i}^{2} (1 ≤ i ≤ r) and t_{i} = τ_{i} (r + 1 ≤ i ≤ 3g − 3).

From now on we specialize to the case of even spin curves and describe the boundary of \overline{S}_{g}^{+}. In the process we determine the ramification of the finite covering π : \overline{S}_{g}^{+} → \overline{M}_{g}.

1.1. The boundary divisors of \overline{S}_{g}^{+}.

If [X, η, β] ∈ π^{−1}([C ∪_{y} D]) where [C, y] ∈ M_{i,1} and [D, y] ∈ M_{g−i,1}, then necessarily X := C ∪_{y_{1}} E ∪_{y_{2}} D, where E is an exceptional component such that C ∩E = {y_{1}} and D ∩ E = {y_{2}}. Moreover

η = (η_{C}, η_{D}, η_{E} = O_{E}(1)) ∈ Pic^{g−1}(X),

where η_{C}^{⊗2} = K_{C}, η_{D}^{⊗2} = K_{D}. The condition h^{0}(X, η) ≡ 0 mod 2, implies that the thetacharacteristics η_{C} and η_{D} have the same parity. We denote by A_{i} ⊂ \overline{S}_{g}^{+} the closure of the locus corresponding to pairs ([C, y, η_{C}], [D, y, η_{D}]) ∈ S_{i,1}^{+} × S_{g−i,1}^{+} and by B_{i} ⊂ \overline{S}_{g}^{+} the closure of the locus corresponding to pairs ([C, y, η_{C}], [D, y, η_{D}]) ∈ S_{i,1}^{−} × S_{g−i,1}^{−}.

For a general point [X, η, β] ∈ A_{i}∪B_{i} we have that Aut_{0}(X, η, β) = Aut(X, η, β) = ℤ_{2}. Using (1), the map ℂ_{τ}^{3g−3} → ℂ_{t}^{3g−3} is given by t_{1} = τ_{1}^{2} and t_{i} = τ_{i} for i ≥ 2. Furthermore, Aut_{0}(X, η, β) acts on ℂ_{τ}^{3g−3} via (τ_{1}, τ_{2}, …, τ_{3g−3}) ↦ (−τ_{1}, τ_{2}, …, τ_{3g−3}). It follows that Δ_{i} ⊂ \overline{M}_{g} is not a branch divisor for π : \overline{S}_{g}^{+} → \overline{M}_{g} and if α_{i} = [A_{i}] ∈ Pic(\overline{S}_{g}^{+}) and β_{i} = [B_{i}] ∈ Pic(\overline{S}_{g}^{+}), then for 1 ≤ i ≤ [g/2] we have the relation

(2)  π^{∗}(δ_{i}) = α_{i} + β_{i}.

Moreover, π_{∗}(α_{i}) = 2^{g−2}(2^{i} + 1)(2^{g−i} + 1)δ_{i} and π_{∗}(β_{i}) = 2^{g−2}(2^{i} − 1)(2^{g−i} − 1)δ_{i}.

For a point [X, η, β] such that st(X) = C_{yq} := C/y ∼ q, with [C, y, q] ∈ M_{g−1,2}, there are two possibilities depending on whether X possesses an exceptional component or not. If X = C_{yq} and η_{C} := ν^{∗}(η) where ν : C → X denotes the normalization map, then η_{C}^{⊗2} = K_{C}(y + q). For each choice of η_{C} ∈ Pic^{g−1}(C) as above, there is precisely one choice of gluing the fibres η_{C}(y) and η_{C}(q) such that h^{0}(X, η) ≡ 0 mod 2. We denote by A_{0} the closure in \overline{S}_{g}^{+} of the locus of points [C_{yq}, η_{C} ∈ \sqrt{K_{C}(y + q)}] as above and clearly deg(A_{0}/Δ_{0}) = 2^{2g−2}.

If X = C ∪_{{y,q}} E where E is an exceptional component, then η_{C} := η ⊗ O_{C} is a theta-characteristic on C. Since H^{0}(X, ω) ≅ H^{0}(C, ω_{C}), it follows that [C, η_{C}] ∈ S_{g−1}^{+}. For [C, y, q] ∈ M_{g−1,2} 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_{0} ⊂ \overline{S}_{g}^{+} the closure of the locus of points [C ∪_{\{y,q\}} E, η_{C} ∈ \sqrt{K_{C}}, η_{E} = O_{E}(1)]. If α_{0} = [A_{0}] ∈ Pic(\overline{S}_{g}^{+}) and β_{0} = [B_{0}] ∈ Pic(\overline{S}_{g}^{+}), we then have the relation

(3)  π^{∗}(δ_{0}) = α_{0} + 2β_{0}.

Note that π_{∗}(α_{0}) = 2^{2g−2}δ_{0} and π_{∗}(β_{0}) = 2^{g−2}(2^{g−1} + 1)δ_{0}.

1.2. The uniruledness of \overline{S}_{g}^{+} for small g.

We employ a simple negativity argument to determine κ(\overline{S}_{g}^{+}) for small genus. Using an analogous idea we showed that similarly, for the moduli space of Prym curves, one has that κ(\overline{R}_{g}) = −∞ for g < 8, cf. [FL] Theorem 0.7.

Theorem 1.2. For g < 8, the space \overline{S}_{g}^{+} is uniruled. Proof. We start with a fixed K3 surface S carrying a Lefschetz pencil of curves of genus g. This induces a fibration f : Bl_{g^{2}}(S) → P^{1} and then we set B := (m_{f})_{∗}(P^{1}) ⊂ \overline{M}_{g}, where m_{f} : P^{1} → \overline{M}_{g} is the moduli map m_{f}(t) := [f^{−1}(t)]. We have the following well-known formulas on \overline{M}_{g} (cf. [FP] Lemma 2.4):

B · λ = g + 1, B · δ_{0} = 6g + 18, and B · δ_{i} = 0 for i ≥ 1.

We lift B to a pencil R ⊂ \overline{S}_{g}^{+} of spin curves by taking

R := B ×_{\overline{M}_{g}} \overline{S}_{g}^{+} = \{[C_{t}, η_{C_{t}}] ∈ \overline{S}_{g}^{+} : [C_{t}] ∈ B, η_{C_{t}} ∈ \overline{Pic}^{g−1}(C_{t}), t ∈ P^{1}\} ⊂ \overline{S}_{g}^{+}.

Using (3) one computes the intersection numbers with the generators of Pic(\overline{S}_{g}^{+}):

R · λ = (g + 1)2^{g−1}(2^{g} + 1), R · α_{0} = (6g + 18)2^{2g−2} and R · β_{0} = (6g + 18)2^{g−2}(2^{g−1} + 1).

Furthermore, R is disjoint from all the remaining boundary classes of \overline{S}_{g}^{+}, that is, R·α_{i} = R · β_{i} = 0 for 1 ≤ i ≤ [g/2]. One verifies that R · K\overline{S}{+ \atop g} < 0 precisely when g ≤ 7. Since R

is a covering curve for \overline{S}_{g}^{+} in the range g ≤ 7, we find that K\overline{S}{+ \atop g} is not pseudo-effective,

that is, K\overline{S}{+ \atop g} ∈ \overline{Eff}(\overline{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 \overline{S}_{g}^{+} lies outside the pseudo-effective cone as well. One can apply [BDPP] Corollary 0.3, to conclude that \overline{S}_{g}^{+} is uniruled for g ≤ 7. □

2. The geometry of the divisor \overline{Θ}_{null}

We compute the class of the divisor \overline{Θ}_{null} using 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 series 𝔤_{d}^{r} (see also Remark 2.1).

For g ≥ 9, Harer [H] has showed that H^{2}(S_{g}^{+}, ℚ) ≅ ℚ. The range for which this result holds has been recently improved to g ≥ 5 in [P]. In particular, it follows that Pic(\overline{S}_{g}^{+})_{ℚ} is generated by the classes λ, α_{i}, β_{i} for i = 0, …, [g/2]. Thus we can expand the divisor class \overline{Θ}_{null} in terms of the generators of the Picard group

(4)  \overline{Θ}_{null} ≡ λ̄ · λ − ᾱ_{0} · α_{0} − β̄_{0} · β_{0} − ∑_{i=1}^{[g/2]}(ᾱ_{i} · α_{i} + β̄_{i} · β_{i}) ∈ Pic(\overline{S}_{g}^{+})_{ℚ},

and determine the coefficients λ̄, ᾱ_{0}, β̄_{0}, ᾱ_{i} and β̄_{i} ∈ ℚ for 1 ≤ i ≤ [g/2].

Remark 2.1. To show that the class [Θ_{null}] ∈ Pic(S_{g}^{+})_{ℚ} is a multiple of λ and thus, the expansion (4) makes sense for all g ≥ 3, one does not need to know that Pic(S_{g}^{+})_{ℚ} is infinite cyclic. For instance, for even g = 2k − 2 ≥ 4, we note that, via the base point free pencil trick, [C, η] ∈ Θ_{null} if and only if the multiplication map

μ_{C}(A, η) : H^{0}(C, A) ⊗ H^{0}(C, A ⊗ η) → H^{0}(C, A^{⊗2} ⊗ η)

is not an isomorphism for a base point free pencil A ∈ W_{k}^{1}(C). We set M̃_{g} to be the open subvariety consisting of curves [C] ∈ M_{g} such that W_{k−1}^{1}(C) = ∅ and denote by σ : 𝔊_{k}^{1} → M̃_{g} the Hurwitz scheme of pencils 𝔤_{k}^{1} and by

τ : 𝔊_{k}^{1} ×M̃_{g} S_{g}^{+} → S_{g}^{+}, u : 𝔊_{k}^{1} ×M̃_{g} S_{g}^{+} → 𝔊_{k}^{1}

the (generically finite) projections. Then Θ_{null} = τ_{∗}(Z), where

Z = {[A, C, η] ∈ 𝔊_{k}^{1} ×M̃_{g} S_{g}^{+} : μ_{C}(A, η) is not injective}.

Via this determinantal presentation, the class of the divisor Z is expressible as a combination of τ^{∗}(λ), u^{∗}(𝔞), u^{∗}(𝔟), where 𝔞, 𝔟 ∈ Pic(𝔊_{k}^{1})_{ℚ} are the tautological classes defined in e.g. [FL] p.15. Since τ_{∗}(u^{∗}(𝔞)) = π^{∗}(σ_{∗}(𝔞)) (and similarly for the class 𝔟), the conclusion follows. For odd genus g = 2k − 1, one uses a similar argument replacing 𝔊_{k}^{1} with any generically finite covering of M_{g} given by a Hurwitz scheme (for instance, we take the space of pencils 𝔤_{k+1}^{1} with a triple ramification point).

We start the proof of Theorem 0.2 by determining the coefficients of α_{i} and β_{i} (i ≥ 1) in the expansion of [\overline{Θ}_{null}].

Theorem 2.2. We fix integers g ≥ 3 and 1 ≤ i ≤ [g/2]. The coefficient of α_{i} in the expansion of [\overline{Θ}_{null}] equals 0, while the coefficient of β_{i} equals −1/2. That is, ᾱ_{i} = 0 and β̄_{i} = 1/2.

Proof. For each integer 2 ≤ i ≤ g−1, we fix general curves [C] ∈ M_{i} and [D, q] ∈ M_{g−i,1} and consider the test curve C^{i} := \{C ∪_{y∼q} D\}_{y∈C} ⊂ Δ_{i} ⊂ \overline{M}_{g}. We lift C^{i} to test curves F_{i} ⊂ A_{i} and G_{i} ⊂ B_{i} inside \overline{S}_{g}^{+} constructed as follows. We fix even (resp. odd) theta-characteristics η_{C}^{+} ∈ Pic^{i−1}(C) and η_{D}^{+} ∈ Pic^{g−i−1}(D) (resp. η_{C}^{−} ∈ Pic^{i−1}(C) and η_{D}^{−} ∈ Pic^{g−i−1}(D)).

If E ≅ P^{1} is an exceptional component, we define the family F_{i} (resp. G_{i}) as consisting of spin curves

F_{i} := \{t := [C ∪_{y} E ∪_{q} D, η_{C} = η_{C}^{+}, η_{E} = O_{E}(1), η_{D} = η_{D}^{+}] ∈ \overline{S}_{g}^{+} : y ∈ C\}

and

G_{i} := \{t := [C ∪_{y} E ∪_{q} D, η_{C} = η_{C}^{−}, η_{E} = O_{E}(1), η_{D} = η_{D}^{−}] ∈ \overline{S}_{g}^{+} : y ∈ C\}.

Since π_{∗}(F_{i}) = π_{∗}(G_{i}) = C^{i}, clearly F_{i} · α_{i} = C^{i} · δ_{i} = 2 − 2i, F_{i} · β_{i} = 0 and F_{i} has intersection number 0 with all other generators of Pic(\overline{S}_{g}^{+}). Similarly

G_{i} · β_{i} = 2 − 2i, G_{i} · α_{i} = 0, G_{i} · λ = 0,

and G_{i} does not intersect the remaining boundary classes in \overline{S}_{g}^{+}.

Next we determine F_{i} ∩ \overline{Θ}_{null}. Assume that a point t ∈ F_{i} lies in \overline{Θ}_{null}. Then there exists a family of even spin curves (f : X → S, η, β), where S = Spec(R), with R being a discrete valuation ring and X is a smooth surface, such that, if 0, ξ ∈ S denote the special and the generic point of S respectively and X_{ξ} is the generic fibre of f, then

h^{0}(X_{ξ}, η_{ξ}) ≥ 2, h^{0}(X_{ξ}, η_{ξ}) ≡ 0 mod 2, η_{ξ}^{⊗2} ≅ ω_{X_{ξ}} and (f^{−1}(0), η_{f^{−1}(0)}) = t ∈ \overline{S}_{g}^{+}.

Following the procedure described in [EH1] p. 347-351, this data produces a limit linear series 𝔤_{g−1}^{1} on C ∪ D, say

l := (l_{C} = (L_{C}, V_{C}), l_{D} = (L_{D}, V_{D})) ∈ G_{g−1}^{1}(C) × G_{g−1}^{1}(D),

such that the underlying line bundles L_{C} and L_{D} respectively, are obtained from the line 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. For degree reasons, it follows that L_{C} = η_{C}^{+} ⊗ O_{C}((g − i)y) and L_{D} = η_{D}^{+} ⊗ O_{D}(iq). Since both C and D are general in their respective moduli spaces, we have that H^{0}(C, η_{C}^{+}) = 0 and H^{0}(D, η_{D}^{+}) = 0. In particular a_{1}^{l_{C}}(y) ≤ g − i − 1 and a_{0}^{l_{D}}(q) < a_{1}^{l_{D}}(q) ≤ i − 1, hence a_{1}^{l_{C}}(y) + a_{0}^{l_{D}}(q) ≤ g − 2, which contradicts the definition of a limit 𝔤_{g−1}^{1}. Thus F_{i} ∩ \overline{Θ}_{null} = ∅. This implies that ᾱ_{i} = 0, for all 1 ≤ i ≤ [g/2] (for i = 1, one uses instead the curve F_{g−1} ⊂ A_{1} to reach the same conclusion).

Assume that t ∈ G_{i} ∩ \overline{Θ}_{null}. By the same argument as above, retaining also the notation, there is an induced limit linear series on C ∪ D,

(l_{C}, l_{D}) ∈ G_{g−1}^{1}(C) × G_{g−1}^{1}(D),

where 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,1} are both general, we may assume that h^{0}(D, η_{D}^{−}) = h^{0}(C, η_{C}^{−}) = 1, q ∉ supp(η_{D}^{−}) and that supp(η_{C}^{−}) consists of i − 1 distinct points. In particular a_{1}^{l_{D}}(q) ≤ i, hence a_{0}^{l_{C}}(y) ≥ g − 1 − a_{1}^{l_{D}}(q) ≥ g − i − 1. Since h^{0}(C, η_{C}^{−}) = 1, it follows that one has in fact equality, that is, a_{0}^{l_{C}}(y) = g − i − 1 and then necessarily a_{1}^{l_{D}}(q) = i.

Similarly, a_{1}^{l_{C}}(y) ≤ g − i + 1 (otherwise div(η_{C}^{−}) ≥ 2y, that is, supp(η_{C}^{−}) would be non-reduced, a contradiction), thus a_{0}^{l_{D}}(q) ≥ i − 2, and the last two inequalities must be equalities as well (one uses that h^{0}(D, L_{D} ⊗ O_{D}(−(i − 1)q)) = h^{0}(D, η_{D}^{−} ⊗ O_{D}(q)) = 1, that is, a_{0}^{l_{D}}(q) < i − 1). Since a_{1}^{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−1}^{1} and in fact

(5) l_{D} = |η_{D}^{−}⊗O_{D}(2q)|+(i−2)·q ∈ G_{g−1}^{1}(D), l_{C} = |η_{C}^{−}⊗O_{C}(y)|+(g−i−1)·y ∈ G_{g−1}^{1}(C),

hence a^{l_{D}}(q) = (i − 2, i) and a^{l_{C}}(y) = (g − i − 1, g − i + 1).

To prove that the intersection between G_{i} and \overline{Θ}_{null} is transversal, we follow closely [EH3] Lemma 3.4 (see especially the Remark on p. 45): The restriction \overline{Θ}_{null |G_{i}} is isomorphic, as a scheme, to the variety τ : 𝔗_{g−1}^{1}(G_{i}) → G_{i} of limit linear series 𝔤_{g−1}^{1} on the curves of compact type {C ∪_{y∼q} D : y ∈ C}, whose C and D-aspects are obtained by twisting suitably at y ∈ C and q ∈ D the fixed theta-characteristics η_{C}^{−} and η_{D}^{−} respectively. 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 of singular spin curves of compact type, the scheme 𝔗_{g−1}^{1}(G_{i}) splits as a product of the corresponding moduli spaces of C and D-aspects respectively of the limits 𝔤_{g−1}^{1}. By direct calculation we have showed that 𝔗_{g−1}^{1}(G_{i}) ≅ supp(η_{C}^{−}) × {l_{D}}. Since supp(η_{C}^{−}) is a reduced 0-dimensional scheme, we obtain that \overline{Θ}_{null |G_{i}} is everywhere reduced. It follows that G_{i} · \overline{Θ}_{null} = #supp(η_{C}^{−}) = i − 1 and then β̄_{i} = (G_{i} · \overline{Θ}_{null})/(2i − 2). This argument does not work for i = 1, when one uses instead the intersection of \overline{Θ}_{null} with G_{g−1}, and this finishes the proof. □

Next we construct two pencils in \overline{S}_{g}^{+} which are lifts of the standard degree 12 pencil of elliptic tails in \overline{M}_{g}. We fix a general pointed curve [C, q] ∈ M_{g−1,1} and a pencil f : Bl_{9}(P^{2}) → P^{1} of plane cubics together with a section σ : P^{1} → Bl_{9}(P^{2}) induced by one of the base points. We then consider the pencil R := \{[C∪_{q∼σ(λ)}f^{−1}(λ)]\}_{λ∈P^{1}} ⊂ \overline{M}_{g}.

We fix an odd theta-characteristic η_{C}^{−} ∈ Pic^{g−2}(C) such that q ∉ supp(η_{C}^{−}) and E ≅ P^{1} will again denote an exceptional component. We define the family

F_{0} := \{[C ∪_{q} E ∪_{σ(λ)} f^{−1}(λ), η_{C} = η_{C}^{−}, η_{E} = O_{E}(1), η_{f^{−1}(λ)} = O_{f^{−1}(λ)}] : λ ∈ P^{1}\} ⊂ \overline{S}_{g}^{+}.

Since F_{0} ∩A_{1} = ∅, we find that F_{0} ·β_{1} = π_{∗}(F_{0})·δ_{1} = −1. Similarly, F_{0} ·λ = π_{∗}(F_{0})·λ = 1 and obviously F_{0} · α_{i} = F_{0} · β_{i} = 0 for 2 ≤ i ≤ [g/2]. For each of the 12 points λ_{∞} ∈ P^{1} corresponding to singular fibres of R, the associated η_{λ_{∞}} ∈ \overline{Pic}^{g−1}(C ∪ E ∪ f^{−1}(λ_{∞})) are actual line bundles on C ∪E ∪f^{−1}(λ_{∞}) (that is, we do not have to blow-up the extra node). Thus we obtain that F_{0} · β_{0} = 0, therefore F_{0} · α_{0} = π_{∗}(F_{0}) · δ_{0} = 12.

We also fix an even theta-characteristic η_{C}^{+} ∈ Pic^{g−2}(C) and consider the degree 3 branched covering γ : \overline{S}_{1,1}^{+} → \overline{M}_{1,1} forgetting the spin structure. We define the pencil

G_{0} := \{[C∪_{q}E∪_{σ(λ)}f^{−1}(λ), η_{C} = η_{C}^{+}, η_{E} = O_{E}(1), η_{f^{−}1(λ)} ∈ γ^{−1}[f^{−1}(λ)]] : λ ∈ P^{1}\} ⊂ \overline{S}_{g}^{+}.

Since π_{∗}(G_{0}) = 3R, we have that G_{0} · λ = 3. Obviously G_{0} · β_{0} = G_{0} · β_{1} = 0, hence G_{0} · α_{1} = π_{∗}(G_{0}) · δ_{1} = −3. The map γ : \overline{S}_{1,1}^{+} → \overline{M}_{1,1} is simply ramified over the point corresponding to j-invariant ∞. Hence, G_{0} · α_{0} = 12 and G_{0} · β_{0} = 12, which is consistent with formula (3).

The last pencil we construct lies in the boundary divisor B_{0} ⊂ \overline{S}_{g}^{+}: Setting E ≅ P^{1} for an exceptional component, we define

H_{0} := \{[C ∪_{\{y,q\}} E, η_{C} = η_{C}^{+}, η_{E} = O_{E}(1)] : y ∈ C\} ⊂ \overline{S}_{g}^{+}.

The fibre of H_{0} over the point y = q ∈ C is the even spin curve

[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′′}(−1)],

having as stable model [C ∪_{q} E_{∞}], where E_{∞} := E′′/y′′ ∼ q′′ is the rational nodal curve corresponding 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_{0}·λ = 0, H_{0}·α_{i} = H_{0}·β_{i} = 0 for 2 ≤ i ≤ [g/2]. Moreover H_{0}·α_{0} = 0, hence H_{0} · β_{0} = \frac{1}{2}π_{∗}(H_{0}) · δ_{0} = 1 − g. Finally, H_{0} · α_{1} = 1 and H_{0} · β_{1} = 0.

Theorem 2.3. If F_{0}, G_{0}, H_{0} ⊂ \overline{S}_{g}^{+} are the families of spin curves defined above, then

F_{0} · \overline{Θ}_{null} = G_{0} · \overline{Θ}_{null} = H_{0} · \overline{Θ}_{null} = 0.

Proof. From the limit linear series argument in the proof of Theorem 2.2 we get that the assumption F_{0} ∩ \overline{Θ}_{null} ≠ ∅ implies that q ∈ supp(η_{C}^{−}), a contradiction. Similarly, we have that G_{0} ∩ \overline{Θ}_{null} = ∅ because [C] ∈ M_{g−1} can be assumed to have no even theta-characteristics η_{C}^{+} ∈ Pic^{g−2}(C) with h^{0}(C, η_{C}^{+}) ≥ 2, that is [C, η_{C}^{+}] ∉ \overline{Θ}_{null} ⊂ \overline{S}_{g−1}^{+}. Finally, we assume that there exists a point [X := C ∪_{{y,q}} E, η_{C} = η_{C}^{+}, η_{E} = O_{E}(1)] ∈ H_{0} ∩ \overline{Θ}_{null}. Then certainly h^{0}(X, η_{X}) ≥ 2 and from the Mayer-Vietoris sequence on X we find that

H^{0}(X, η_{X}) = Ker{H^{0}(C, η_{C}) ⊕ H^{0}(E, O_{E}(1)) → ℂ_{y,q}^{2}},

hence h^{0}(C, η_{C}) = h^{0}(X, η_{X}) ≥ 2. This contradicts the assumption that [C] ∈ M_{g−1} is general. A similar argument works for the special point in H_{0} ∩ π^{−1}(Δ_{1}), hence H_{0} · \overline{Θ}_{null} = 0. □

Proof of Theorem 0.2. Looking at the expansion of [\overline{Θ}_{null}], Theorem 2.3 gives the relations

F_{0} · \overline{Θ}_{null} = λ̄ − 12ᾱ_{0} + β̄_{1} = 0, G_{0} · \overline{Θ}_{null} = 3λ̄ − 12ᾱ_{0} − 12β̄_{0} + 3ᾱ_{1} = 0
and H_{0} · \overline{Θ}_{null} = (g − 1)β̄_{0} − ᾱ_{1} = 0.

Since we have already computed ᾱ_{i} = 0 and β̄_{i} = 1/2 for 1 ≤ i ≤ [g/2], (cf. Theorem 2.2), we obtain that λ̄ = 1/4, ᾱ_{0} = 1/16 and β̄_{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 M_{g}^{1} is the locus of curves [C] ∈ M_{g} with a vanishing theta-null then its closure has class equal to

\overline{M}_{g}^{1} ≡ 2^{g−3}((2^{g} + 1)λ − 2^{g−3}δ_{0} − ∑_{i=1}^{[g/2]}(2^{g−i} − 1)(2^{i} − 1)δ_{i}) ∈ Pic(\overline{M}_{g}).

Proof. We use the scheme-theoretic equality π_{∗}(\overline{Θ}_{null}) = \overline{M}_{g}^{1} as well as the formulas π_{∗}(λ) = 2^{g−1}(2^{g} +1)λ, π_{∗}(α_{0}) = 2^{2g−2}δ_{0}, π_{∗}(β_{0}) = 2^{g−2}(2^{g−1} +1)δ_{0}, π_{∗}(α_{i}) = 2^{g−2}(2^{i} + 1)(2^{g−i} + 1)δ_{i} and π_{∗}(β_{i}) = 2^{g−2}(2^{i} − 1)(2^{g−i} − 1)δ_{i} valid for 1 ≤ i ≤ [g/2]. □

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Humboldt-Universität zu Berlin, Institut Für Mathematik, 10099 Berlin

      E-mail address: farkas@math.hu-berlin.de