THE GEOMETRY OF THE MODULI SPACE OF ODD SPIN CURVES

GAVRIL FARKAS AND ALESSANDRO VERRA

The set of odd theta-characteristics on a general curve C of genus g is in bijection with the set θ(C) of theta hyperplanes H ∈ (P^{g−1})^{∨} everywhere tangent to the canoni-

cally embedded curve C ↪^{|K_{C}|} P^{g−1}. Even though the geometry and the intricate combinatorics of θ(C) have been studied classically, see [Dol], [DK] for a modern account, it was only recently proved in [CS] that one can reconstruct a general curve [C] ∈ M_{g} from the hyperplane configuration θ(C).

Odd theta-characteristics form a moduli space π : S_{g}^{−} → M_{g} which is an étale cover of degree 2^{g−1}(2^{g} − 1). The normalization of \overline{M}_{g} in the function field of S_{g}^{−} gives rise to a finite covering π : \overline{S}_{g}^{−} → \overline{M}_{g}. Furthermore, \overline{S}_{g}^{−} has a modular meaning being isomorphic to the coarse moduli space of the Deligne-Mumford stack of odd stable spin curves, cf. [C], [CCC], [AJ]. The map π is branched along the boundary of \overline{M}_{g} and one expects K\overline{S}{− \atop g} to enjoy better positivity properties than K_{\overline{M}_{g}}.

The aim of this paper is to describe the birational geometry of \overline{S}_{g}^{−} for all g. Our goals are (1) to understand the transition from rationality to maximal Kodaira dimension for \overline{S}_{g}^{−} as g increases, and (2) to use the existence of Mukai models of \overline{M}_{g} in order to construct explicit unirational parameterizations of \overline{S}_{g}^{−} for small genus. Remarkably, we end up having no gaps in the classification of \overline{S}_{g}^{−}. First, we show that in the range where the general curve [C] ∈ M_{g} lies on a K3 surface, the existence of special theta pencils on K3 surfaces, provides an explicit uniruled parameterization of \overline{S}_{g}^{−}:

Theorem 0.1. The odd spin moduli space \overline{S}_{g}^{−} is uniruled for g ≤ 11.

When g ≤ 9 or g = 11, a general spin curve [C, η] ∈ \overline{S}_{g}^{−} appears as a hyperplane section of a K3 surface X ⊂ P^{g}, such that if d := supp(η) is the support of the theta-characteristic, then the linear span ⟨d⟩ ⊂ P^{g} is a codimension 2 linear subspace. A rational curve P ⊂ \overline{S}_{g}^{−} is induced by the pencil of hyperplanes PH^{0}(X, I_{d/X}(C)) containing ⟨d⟩. We show in Section 3 that P ⊂ \overline{S}_{g}^{−} is a covering rational curve, satisfying

P · K\overline{S}{− \atop g} = 2g − 24 < 0.

Thus P · K\overline{S}{− \atop g} < 0 precisely when g ≤ 11, which highlights the fact that the nature of \overline{S}_{g}^{−} is expected to change exactly when g ≥ 12. This is something we shall achieve in the course of proving Theorem 0.3.

The previous argument no longer works for \overline{S}_{10}^{−}, when the condition that a curve [C] ∈ \overline{M}_{10} lie on a K3 surface is divisorial [FP]. This case is in some sense a specialization of the genus 11 case. We use that a general 1-nodal irreducible curve [C] ∈ Δ_{0} ⊂ \overline{M}_{11} of arithmetic genus 11, lies on a K3 surface X ⊂ P^{11}. By a degeneration argument, we show that this construction can be also carried out in such a way, that if ν : C′ → C denotes the normalization of C, then the points x, y ∈ C′ with ν(x) = ν(y) (that is, mapping to the node of C), lie in the support of one of the odd-theta characteristics of [C′] ∈ M_{10}. Ultimately, this produces a rational curve P ⊂ \overline{S}_{10}^{−} through a general point, which shows that \overline{S}_{10}^{−} is uniruled as well.

In the range in which a Mukai model of \overline{M}_{g} exists, our results are more precise:

Theorem 0.2. \overline{S}_{g}^{−} is unirational for g ≤ 8.

The proof relies on the existence, in this range, of Mukai varieties V_{g} ⊂ P^{n_{g}+g−2}, where n_{g} = dim(V_{g}), which have the property that general 1-dimensional linear sections of V_{g} are canonical curves [C] ∈ M_{g} with general moduli. We fix an integer 1 ≤ δ ≤ g − 1 and consider the correspondence

P_{g,δ}^{o} := {(C, Γ, Z) : Z ⊂ C ∩ Γ ⊂ V_{g}, |sing(Γ)| = δ, sing(Γ) ⊂ Z},

where Z ⊂ V_{g} is a 0-dimensional subscheme of V_{g} of length 2g − 2, supported at g − 1 points and such that dim⟨Z⟩ = g − 2 (see Section 4 for a precise definition), Γ ⊂ V_{g} is an irreducible δ-nodal curve section of V_{g} whose nodes are among the points in the support of Z, and C ⊂ V_{g} is an arbitrary curve linear section of V_{g} containing Z as a subscheme. Thus if C is smooth, then Z ⊂ C is a divisor of even degree at each point in its support, and O_{C}(Z/2) can be viewed as a theta-characteristic. The variety P_{g,δ}^{o} comes equipped with two projections

\overline{S}_{g}^{−} ⟵^{α} P_{g,δ}^{o} ⟶^{β} B_{g,δ}^{−},

where B_{g,δ}^{−} ⊂ \overline{S}_{g}^{−} denotes the moduli space of irreducible δ-nodal curves of arithmetic genus g together with an odd theta- characteristic on the normalization. It is easy to see that P_{g,δ}^{o} is birational to a projective bundle over the irreducible variety B_{g,δ}^{−}. Thus the

unirationality of \overline{S}_{g}^{−} follows once we prove that (i) α is dominant, and (ii) B_{g,δ}^{−} itself is unirational. We carry out this program when g ≤ 8. In the process of proving Theorem 0.2, we establish some facts of independent interest concerning the Mukai models

𝔐_{g} := G(g, n_{g} + g − 1)^{ss}//Aut(V_{g}).

These are birational models of \overline{M}_{g} having Pic(𝔐_{g}) = ℤ and appearing as GIT quotients of Grassmannians; they can be viewed as log-minimal models of \overline{M}_{g} emerging from the constructions carried out in [M1], [M2], [M3].

Theorem 0.1 is sharp and the remaining moduli spaces \overline{S}_{g}^{−} are of general type:

Theorem 0.3. The space \overline{S}_{g}^{−} is a variety of general type for g > 11.

The border case of \overline{S}_{12}^{−} is particularly challenging and takes up the entire Section 6. We remark that in the range 11 < g < 17, of the two moduli spaces \overline{S}_{g}^{−} and \overline{M}_{g}, one is of general type whereas the other has negative Kodaira dimension. More strikingly, Theorems 0.3 and 0.1 coupled with results from [F3], show that for 9 ≤ g ≤ 11, the space \overline{S}_{g}^{−} is uniruled while \overline{S}_{g}^{+} is of general type! Finally, we note that \overline{S}_{8}^{−} is unirational whereas \overline{S}_{8}^{+} is of Calabi-Yau type [FV].

We describe the main steps in the proof of Theorem 0.3. First, we use that for all g ≥ 4 and l ≥ 0, if ϵ : Ŝ_{g} → \overline{S}_{g}^{−} denotes a resolution of singularities, then there is an induced isomorphism, see [Lud]

ϵ^{∗} : H^{0}(\overline{S}_{g,reg}^{−}, K_{\overline{S}_{g}^{−}}^{⊗l}) →^{∼} H^{0}(Ŝ_{g}, K⊗l ; Ŝ_{g}).

Thus to conclude that \overline{S}_{g}^{−} is of general type, it suffices to exhibit an effective divisor D on \overline{S}_{g}^{−} such that for appropriately chosen rational constants α, β > 0, a relation of the type K\overline{S}{− \atop g} ≡ α λ + β D + E ∈ Pic(\overline{S}_{g}^{−}) holds, where λ ∈ Pic(\overline{S}_{g}^{−}) is the pull-back to \overline{S}_{g}^{−} of the Hodge class, and E is an effective ℚ-class which is typically a combination of boundary divisors. It is essential to pick D so that (1) its class can be explicitly computed, that is, points in D have good geometric characterization, and (2) [D] ∈ Pic(\overline{S}_{g}^{−}) is in some way an extremal point of the effective cone of divisors so that the coefficients α, β stand a chance of being positive. In the case of \overline{S}_{g}^{+}, the role of D is played by the divisor \overline{Θ}_{null} of vanishing theta-nulls, see [F3]. In the case of \overline{S}_{g}^{−} we compute the class of degenerate theta-characteristics, that is, curves carrying a non-reduced odd theta-characteristic.

Theorem 0.4. We fix g ≥ 3. The locus consisting of odd spin curves

Z_{g} := {[C, η] ∈ S_{g}^{−} : η = O_{C}(2x_{1} + x_{2} + ⋯ + x_{g−2}) wherex_{i} ∈ C fori = 1, …, g − 2}

is a divisor on S_{g}^{−}. The class of its compactification inside \overline{S}_{g}^{−} equals

\overline{Z}_{g} ≡ (g + 8)λ − \frac{g + 2}{4}α_{0} − 2β_{0} − ∑_{i=1}^{[g/2]} 2(g − i) α_{i} − ∑_{i=1}^{[g/2]} 2i β_{i} ∈ Pic(\overline{S}_{g}^{−}),

where λ, α_{0}, β_{0}, …, α_{[g/2]}, β_{[g/2]} are the standard generators of Pic(\overline{S}_{g}^{−}).

For low genus, Z_{g} specializes to well-known geometric loci. For instance \overline{Z}_{3} is the divisor of hyperflexes on plane quartics. In particular, Theorem 0.4 yields the formula

π_{∗}(\overline{Z}_{3}) = 308λ − 32δ_{0} − 76δ_{1} ∈ Pic(\overline{M}_{3}),

for the class of quartic curves having a hyperflex. This matches [Cu] formula (5.5). Moreover, one has the following relation in Pic(\overline{M}_{3})

[\{[C] ∈ M_{3} : ∃x ∈ C with 4x ≡ K_{C}\}]^{−} ≡ 8 · \overline{M}_{3,2}^{1} + π_{∗}(\overline{Z}_{3}),

where \overline{M}_{3,2}^{1} ≡ 9λ − δ_{0} − 3δ_{1} is the hyperelliptic class and the multiplicity 8 accounts for the number of hyperelliptic Weierstrass points.

We briefly explain how Theorem 0.4 implies that \overline{S}_{g}^{−} is of general type for g > 11. We choose an effective divisor D ∈ Eff(\overline{M}_{g}) of small slope; for composite g + 1 one can take D = \overline{M}_{g,d}^{r} the closure of the Brill-Noether divisor of curves with a 𝔤_{d}^{r}, where ρ(g, r, d) = −1; there exists a constant c_{g,d,r} > 0 such that [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}).

We form the linear combination of divisors on \overline{S}_{g}^{−}

\frac{2}{g − 2}\overline{Z}_{g} + \frac{3(3g − 10)}{c_{g,d,r}(g − 2)(g + 1)}π^{∗}(\overline{M}_{g,d}^{r}) = \frac{11g + 37}{g + 1}λ − 2α_{0} − 3β_{0} − ∑_{i=1}^{[g/2]}(a_{i} · α_{i} + b_{i} · β_{i}),

where a_{i}, b_{i} ≥ 2 for i ≠ 1 and a_{1}, b_{1} > 3 are explicitly known rational constants. The canonical class of \overline{S}_{g}^{−} is given by the Riemann-Hurwitz formula

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}),

and by comparison, it follows that for g > 12 one can find a constant μ_{g} ∈ ℚ_{>0} such that

K\overline{S}{− \atop g} − μ_{g} · λ ∈ ℚ_{≥0}⟨[\overline{Z}_{g}], α_{1}, β_{1}, …, α_{[g/2]}, β_{[g/2]}⟩,

which shows that K\overline{S}{− \atop g} is big and thus proves Theorem 0.3.

For g = 12, there is no Brill-Noether divisor, and the reasoning above shows that in order to conclude that \overline{S}_{12}^{−} is of general type, one needs an effective divisor \overline{𝔇}_{12} of slope s(\overline{𝔇}_{12}) < 6 + 12/13, that is, a counterexample to the Slope Conjecture. We define

𝔇_{12} := {[C] ∈ M_{12} : ∃L ∈ W_{14}^{4}(C) with Sym^{2}H^{0}(C, L) ⟶^{μ_{0}(L)} H^{0}(C, L^{⊗2}) not injective},

that is, points in 𝔇_{12} correspond to curves that admit an embedding C ⊂ P^{4} with deg(C) = 14, such that H^{0}(P^{4}, I_{C/P^{4}}(2)) ≠ 0. The computation of the class of \overline{𝔇}_{12} ⊂ \overline{M}_{12} is carried out in Section 6 and it turns out that s(\overline{𝔇}_{12}) = \frac{4415}{642} < 6 + \frac{12}{13}. In particular 𝔇_{12} violates the Slope Conjecture on \overline{M}_{12}, and as such, it contains the locus K_{12} := {[C] ∈ M_{12} : C lies on a K3 surface}.

1. Families of stable spin curves

We briefly review some relevant facts about the moduli space \overline{S}_{g}^{−} that will be used throughout the paper, see also [C], [F3], [Lud] for details. As a matter of notation, we follow the convention set in [FL]; if M is a Deligne-Mumford stack, then we denote by M its associated coarse moduli space.

Following [C], 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.

It follows from the definition that if (X, η, β) is a spin curve with exceptional components E_{1}, …, E_{r} and \{p_{i}, q_{i}\} = E_{i} ∩ \overline{X − E_{i}} for i = 1, …, r, then β_{|E_{i}} = 0. Moreover, if X̃ := \overline{X − ⋃_{i=1}^{r} E_{i}} (viewed as a subcurve of X), then we have an isomorphism of

sheaves η⊗2 ; X̃ →^{∼} ωX̃.

We denote by \overline{S}_{g} the non-singular Deligne-Mumford stack of spin curves of genus g, which obviously splits into two connected components \overline{S}_{g}^{+} and \overline{S}_{g}^{−} of relative degree 2^{g−1}(2^{g} + 1) and 2^{g−1}(2^{g} − 1) respectively. It is proved in [C] that the coarse moduli space of \overline{S}_{g} is isomorphic to the normalization of \overline{M}_{g} in the function field of S_{g}. There is a proper morphism π : \overline{S}_{g} → \overline{M}_{g} given by π([X, η, β]) := [st(X)], where st(X) denotes the stable model of the nodal curve X.

1.1. Spin curves of compact type. We recall the description of the pull-back divisors π^{∗}(Δ_{i}). We choose a spin curve [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), and since β_{E} = 0, it follows that η_{C}^{⊗2} = K_{C}, η_{D}^{⊗2} = K_{D}, that is, η_{C} and η_{D} are ”honest” theta-characteristics on C and D respectively. The condition h^{0}(X, η) ≡ 1 mod 2 implies that η_{C} and η_{D} must have opposite parities. We denote by A_{i} ⊂ \overline{S}_{g}^{−} the closure in \overline{S}_{g}^{−} of the locus corresponding to pairs

([C, η_{C}, y], [D, η_{D}, y]) ∈ S_{i,1}^{−} × S_{g−i,1}^{+},

and by B_{i} ⊂ \overline{S}_{g}^{−} the closure in \overline{S}_{g}^{−} of the locus corresponding to pairs

([C, η_{C}, y], [D, η_{D}, y]) ∈ S_{i,1}^{+} × S_{g−i,1}^{−}.

One has the relation π^{∗}(Δ_{i}) = A_{i} + B_{i} and clearly deg(A_{i}/Δ_{i}) = 2^{g−2}(2^{i} − 1)(2^{g−i} + 1) and deg(B_{i}/Δ_{i}) = 2^{g−2}(2^{i} + 1)(2^{g−i} − 1). One denotes α_{i} := [A_{i}], β_{i} := [B_{i}] ∈ Pic(\overline{S}_{g}^{−}).

1.2. Spin curves with an irreducible stable model. In order to describe π^{∗}(Δ_{0}) we pick a point [X, η, β] such that st(X) = C_{yq} := C/y ∼ q, where [C, y, q] ∈ M_{g−1,2} is a general point of Δ_{0}. Unlike the case of curves of compact type, here 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) ≡ 1 mod 2. We denote by A_{0} the closure in \overline{S}_{g}^{−} of the locus of those points [C_{yq}, η_{C} ∈ \sqrt{K_{C}(y + q)}] with η_{C}(y) and η_{C}(q) glued as above. One has that deg(A_{0}/Δ_{0}) = 2^{2g−2}.

If X = C ∪_{{y,q}} E where E is an exceptional component, then since β_{|E} = 0 it follows that β_{|C} ∈ H^{0}(C, ω_{X|C} ⊗ η_{C}^{⊗(−2)}) must vanish at both y and q and then for degree reasons η_{C} := η ⊗ O_{C} is a theta-characteristic on C. The condition H^{0}(X, ω) ≅ H^{0}(C, ω_{C}) ≡ 1 mod 2 implies that [C, η_{C}] ∈ S_{g−1}^{−}. In an étale neighborhood of a point [X, η, β], the covering π is given by

(τ_{1}, τ_{2}, …, τ_{3g−3}) ↦ (τ_{1}^{2}, τ_{2}, …, τ_{3g−3}),

where one identifies ℂ_{τ}^{3g−3} with the versal deformation space of (X, η, β) and the hyperplane (τ_{1} = 0) ⊂ ℂ_{τ}^{3g−3} denotes the locus of spin curves where the exceptional component E persists. This discussion shows that π is simply branched over Δ_{0} and we denote the ramification divisor by B_{0} ⊂ \overline{S}_{g}^{−}, that is, the closure of the locus of spin curves [C ∪_{\{y,q\}} E, (C, η_{C}) ∈ S_{g−1}^{−}, 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

(1)  π^{∗}(δ_{0}) = α_{0} + 2β_{0}.

We define several test curves in the boundary of \overline{S}_{g}^{−} which will be later used to compute divisor classes on the moduli space.

1.3. The family F_{i}. We fix 1 ≤ i ≤ [g/2] and construct a covering family for the boundary divisor A_{i}. We fix general curves [C] ∈ M_{i} and [D, q] ∈ M_{g−i,1} as well as an odd theta-characteristic η_{C}^{−} on C and an even theta-characteristic η_{D}^{+} on D. If E ≅ P^{1} is a fixed exceptional component, we define the family of spin curves

F_{i} := {[C∪_{y}∪E∪_{q}D, η] : η_{C} = η_{C}^{−}, η_{E} = O_{E}(1), η_{D} = η_{D}^{+}, E∩C = {y}, E∩D = {q}}_{y∈C}.

One has that F_{i} · β_{i} = 0 and then F_{i} · α_{i} = −2i + 2; furthermore F_{i} has intersection number zero with the remaining generators of Pic(\overline{S}_{g}^{−}).

1.4. The family G_{i}. As above, we fix 1 ≤ i ≤ [g/2] and curves [C] ∈ M_{i}, [D, q] ∈ M_{g−i,1}. This time we choose an even theta-characteristic η_{C}^{+} on C and an odd thetacharacteristic η_{D}^{−} on D. The following family covers the divisor B_{i}:

G_{i} := {[C∪_{y}∪E∪_{q}D, η] : η_{C} = η_{C}^{+}, η_{E} = O_{E}(1), η_{D} = η_{D}^{−}, E∩C = {y}, E∩D = {q}}_{y∈C}.

Clearly G_{i} · α_{i} = 0, G_{i} · β_{i} = 2 − 2i and G_{i} · λ = G_{i} · α_{j} = G_{i} · β_{j} = 0 for j ≠ i.

1.5. Two elliptic pencils. The boundary divisor Δ_{1} ⊂ \overline{M}_{g} is covered by a standard elliptic pencil R obtained by attaching to a fixed general pointed curve [C, y] ∈ M_{g−1,1} a pencil of plane cubic curves {E_{λ} = f^{−1}(λ)}_{λ∈P^{1}} where f : Bl_{9}(P^{2}) → P^{1}. The points of attachment on the elliptic pencil are given by a section σ : P^{1} → Bl_{9}(P^{2}) given by one of the base points of the pencil of cubics. We lift this pencil in two possible ways to the space \overline{S}_{g}^{−}, depending on the parity of the theta-characteristic on the varying elliptic tail. We fix an even theta-characteristic η_{C}^{+} ∈ Pic^{g−2}(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} ∩B_{1} = ∅, we find that F_{0} ·α_{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 and then F_{0} · α_{0} = π_{∗}(F_{0}) · δ_{0} = 12.

A second lift of the elliptic pencil to \overline{S}_{g}^{−} is obtained by choosing an odd thetacharacteristic η_{C}^{−} ∈ Pic^{g−2}(C) whereas on E_{λ} one takes each of the 3 possible even theta-characteristics, that is,

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}^{−},

where γ : \overline{S}_{1,1}^{+} → \overline{M}_{1,1} is the projection of degree 3. Since π_{∗}(G_{0}) = 3R ⊂ Δ_{1}, we obtain that G_{0} · λ = 3. Obviously G_{0} · α_{1} = 0, hence G_{0} · β_{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.

1.6. A covering family in B_{0}. We start with a general pointed spin curve [C, q, η_{C}^{−}] ∈ S_{g−1,1}^{−} and as usual E ≅ P^{1} denotes an exceptional component. We construct a family of spin curves H_{0} ⊂ B_{0} with general member

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

and with special fibre corresponding to y = q being the odd spin curve with support

C ∪_{q} E′ ∪_{q′} E_{2} ∪_{{y_{2},q_{2}}} E,

where E′ and E_{2} are both smooth rational curves and y_{2}, q_{2} ∈ E, E_{2} ∩ E = {y_{2}, q_{2}}, while E_{2} ∩ E′ = \{q′\}. The stable model of this curve is C ∪_{q} (\frac{E_{2}}{y_{2}∼q_{2}} ), having an elliptic tail of j-invariant ∞. The underlying line bundle η ∈ Pic^{g−1}(C ∪ E′ ∪ E_{2} ∪ E) satisfies η_{C} = η_{C}^{−}, η_{E′} = O_{E′}(1), η_{E} = O_{E}(1) and, for degree reasons, η_{E_{2}} = O_{E_{2}}(−1). We have the following relations for the numerical parameters of H_{0}:

H_{0} · λ = 0, H_{0} · β_{0} = 1 − g, H_{0} · α_{0} = 0, H_{0} · β_{1} = 1, H_{0} · α_{1} = 0.

(The only non-trivial calculation here uses that H_{0} ·β_{0} = π_{∗}(H_{0})·δ_{0}/2 = 1−g, cf. [HM]).

2. The Scorza curve

Here we study in detail the correspondence T_{η} ⊂ C × C associated to each (nonvanishing) theta-characteristic [C, η] ∈ S_{g}^{+} − Θ_{null}. This correspondence was used by G. Scorza [Sc] to provide a birational isomorphism between M_{3} and S_{3}^{+} (see also [DK]), and recently in [TZ] where several conditional statements of Scorza’s have been rigourously established.

For a fixed theta-characteristic [C, η] ∈ S_{g}^{+} − Θ_{null}, we define the curve

T_{η} := {(x, y) ∈ C × C : H^{0}(C, η ⊗ O_{C}(x − y)) ≠ 0}.

By Riemann-Roch, it follows that T_{η} is a symmetric correspondence which misses the diagonal Δ ⊂ C × C. The curve T_{η} has a natural fixed point free involution and we denote by f : T_{η} → Γ_{η} the associated étale double covering. Under the assumption that T_{η} is a reduced curve, its class is computed in [DK] Proposition 7.1.5:

T_{η} ≡ (g − 1)F_{1} + (g − 1)F_{2} + Δ.

Theorem 2.1. For a general theta-characteristic [C, η] ∈ S_{g}^{+}, the Scorza curve T_{η} is a smooth curve of genus g(T_{η}) = 3g(g − 1) + 1.

Proof. It is straightforward to show that a point (x, y) ∈ T_{η} is singular if and only if

(2)  H^{0}(C, η ⊗ O_{C}(x − 2y)) ≠ 0 and H^{0}(C, η ⊗ O_{C}(y − 2x)) ≠ 0.

By induction on g, we show that for a general even spin curve such a pair (x, y) cannot exist. We assume the result holds for a general [C, η_{C}] ∈ S_{g−1}^{+}. We fix a general point q ∈ C, an elliptic curve D together with η_{D} ∈ Pic^{0}(D) − {O_{D}} with η_{D}^{⊗2} = O_{D} and consider the spin curve t := [C ∪ E ∪ D, η_{|C} = η_{C}, η_{|E} = O_{E}(1), η_{|D} = η_{D}] ∈ \overline{S}_{g}^{+}, obtained from C ∪_{q} D by inserting an exceptional component E. Since the exceptional component plays no further role in the proof, we are going to suppress it.

We assume by contradiction that t ∈ \overline{S}_{g}^{+} lies in the closure of the locus of spin curves with singular Scorza curve. Then there exists a nodal curve C ∪_{q} D′ semistably equivalent to C ∪_{q} D obtained by inserting a possibly empty chain on P^{1}’s at the node q (therefore, p_{a}(D′) = 1 and we may regard D as a subcurve of D′), as well as smooth points x, y ∈ C∪D′ together with two limit linear series σ = {σ_{C}, σ_{D′}} and τ = {τ_{C}, τ_{D′}} of type 𝔤_{g−2}^{0} on C ∪ D′ such that the underlying line bundles corresponding to σ (resp. τ) are uniquely determined twists at the nodes of the line bundle η ⊗ O_{C∪D′}(x − 2y) (resp. η ⊗ O_{C∪D′}(y − 2x)). The precise twists are determined by the limit linear series condition that each aspect of a limit 𝔤_{g−2}^{0} have degree g − 2. We distinguish three cases depending on which components of C ∪ D′ the points x and y specialize.

(i) x, y ∈ C. Then σ_{C} ∈ H^{0}(C, η_{C} ⊗ O_{C}(x − 2y + q)), τ_{C} ∈ H^{0}(C, η_{C} ⊗ O_{C}(y − 2x + q)), while σ_{D}, τ_{D} ∈ H^{0}(D, η_{D} ⊗ O_{D}((g − 2)q)). Denoting by \{q′\} ∈ D ∩ \overline{(C ∪ D′) − D} the point where D meets the rest of the curve, one has the compatibility conditions

ord_{q}(σ_{C}) + ord_{q′}(σ_{D}) ≥ g − 2 and ord_{q}(τ_{C}) + ord_{q′}(τ_{D}) ≥ g − 2,

which leads to ord_{q}(σ_{C}) ≥ 1 and ord_{q}(τ_{C}) ≥ 1, that is, we have found two points x, y ∈ C such that H^{0}(C, η_{C}(x − 2y)) ≠ 0 and H^{0}(C, η_{C}(y − 2x)) ≠ 0, which contradicts the inductive assumption on C. (ii) x, y ∈ D′. This case does not appear if we choose η_{C} such that H^{0}(C, η_{C}) = 0. Indeed, for degree reason, both non-zero sections σ_{C}, τ_{C} must lie in the space H^{0}(C, η_{C}). (iii) x ∈ C, y ∈ D′. For simplicity, we assume first that y ∈ D. We find that

σ_{C} ∈ H^{0}(C, η_{C} ⊗ O_{C}(x − q)), σ_{D} ∈ H^{0}(D, η_{D} ⊗ O_{D}(g · q′ − 2y)) and

τ_{C} ∈ H^{0}(C, η_{C} ⊗ O_{C}(2q − 2x)), τ_{D} ∈ H^{0}(D, η_{D} ⊗ O_{C}(y + (g − 3) · q′)). We claim that ord_{q}(σ_{C}) = ord_{q}(τ_{C}) = 0 which can be achieved by a generic choice of q ∈ C. Then ord_{q′}(σ_{D}) ≥ g − 2, which implies that η_{D} = O_{D}(2y − 2q). Similarly, ord_{q}(τ_{D}) ≥ g − 2 which yields that η_{D} = O_{D}(q − y), that is, η_{D}^{⊗3} = O_{D}. Since η_{D} was assumed to be a non-trivial point of order 2 this leads to a contradiction. Finally, the case y ∈ D′ − D, that is, when y lies on an exceptional subcurve E′ ⊂ D′ is dealt with similarly: Since ord_{q}(σ_{C}) = ord_{q}(τ_{C}) = 0, by compatibility, after passing through the component E′, one obtains that ord_{q′}(σ_{D}) ≥ g−2. Since σ_{D} ∈ H^{0}(D, η_{D} ⊗O_{D}((g−2)q′)) and η_{D} ≠ O_{D}, we obtain a contradiction □

3. Theta pencils on K3 surfaces.

In this section we prove Theorem 0.1. As usual, we denote by F_{g} the moduli space of polarized K3 surfaces [X, H], where X is a K3 surface and H ∈ Pic(X) is a (primitive) polarization of degree H^{2} = 2g − 2. For integers 0 ≤ δ ≤ g, we introduce the universal Severi variety of pairs

V_{g,δ} := {([X, H], C) : [X, H] ∈ F_{g} and C ∈ |O_{X}(H)| is an integral δ − nodal curve }.

If σ : V_{g,δ} → F_{g} is the obvious projection, we set V_{g,δ}(|H|) := σ^{−1}([X, H]). It is known that every irreducible component of V_{g,δ} has dimension 19+g−δ and maps dominantly onto F_{g}. It is in general not known whether V_{g,δ} is irreducible, see [De] for interesting work in this direction.

For a point [X, H] ∈ F_{g}, we consider a pencil of curves P ⊂ |H|, and denote by Z the base locus of P. We assume that a general member C ∈ P is a nodal integral curve. It follows that C − Z is smooth and that S := sing(C) is a, possibly empty, subset of Z. Let ϵ : X′ := Bl_{S}(X) → X be the blow-up of X along the locus S of nodes, and denote by E the exceptional divisor of ϵ. Let

P′ ⊂ |ϵ^{∗}H ⊗ O_{X′}(−2E)|

be the strict transform of P by ϵ, and Z′ its base locus. Since a general member C ∈ P is nodal precisely along S, a general curve C′ ∈ P′ is smooth. We view h′ := Z′ + E · C′ as a divisor on the smooth curve C′. By the adjunction formula, h′ ∈ |ω_{C′}|.

Definition 3.1. We say that P is a theta pencil, if h′ has even multiplicity at each of its points, that is, O_{C′}(\frac{1}{2}h′) is an odd theta-characteristic for every smooth curve C′ ∈ P′.

The definition implies that the intersection multiplicity of two curves in P is even at each point p ∈ supp(Z). For every pair [X, H] ∈ F_{g} we have that:

Proposition 3.2. Every smooth curve C ∈ |H| belongs to a theta pencil.

Proof. Let d ∈ C_{g−1} be the support of a theta-characteristic on C such that h^{0}(C, O_{C}(d)) = 1. Then PH^{0}(X, I_{d/X}(H)) is a theta pencil. □

We can reverse the construction of a theta pencil, starting instead with the normalization of a nodal section of a K3 surface. Suppose

t := [C′, x_{1}, y_{1}, …, x_{δ}, y_{δ}, η] ∈ M_{g−δ,2δ} ×_{M_{g−δ}} S_{g−δ}^{−}

is a 2δ-pointed curve together with an isolated odd theta-characteristic η, such that:

(i) h^{0}(C′, η ⊗ O_{C′}(− ∑_{i=1}^{δ}(x_{i} + y_{i}))) ≥ 1; we write supp(η) = ∑_{i=1}^{δ}(x_{i} + y_{i}) + d, where d ∈ C′_{g−3δ−1} is the residual divisor. (ii) There exists a polarized K3 surface [X, H] ∈ F_{g} and a map f : C′ → X, such that f(x_{i}) = f(y_{i}) = p_{i} for all i = 1, …, δ , f_{∗}(C′) ∈ |H|, and moreover f : C′ → C is the normalization map of the δ-nodal curve C := f(C′).

If ϵ : X′ → X is the blow-up of X at the points p_{1}, …, p_{δ} and E := ∑_{i=1}^{δ} E_{p_{i}} ⊂ X′ denotes the exceptional divisor, we may view C′ ⊂ X, where C′ ≡ ϵ^{∗}H − 2E. Then

|I_{d/X′}(C′)| = |I_{2d/X′}(C′)| = |I_{2d+∑_{i=1}^{δ}(x_{i}+y_{i})/X′}(C′)|

is a theta pencil of δ-nodal curves on X.

If K_{g−δ,δ}^{−} ⊂ M_{g−δ,2δ} ×_{M_{g−δ}} S_{g−δ}^{−} is the locus of elements [C, (x_{i}, y_{i})_{i=1,...,δ}, η] satisfying conditions (i) and (ii), the previous discussion proves the following:

Proposition 3.3. Every irreducible component of K_{g−δ,δ}^{−} is uniruled.

This implies the following consequence of Proposition 4.4 to be established in the next section:

Theorem 3.4. We set g ≤ 9 and 0 ≤ δ ≤ (g + 1)/3. Then the variety K_{g−δ,δ}^{−} is non-empty, uniruled and dominates the spin moduli space S_{g−δ}^{−}.

Definition 3.5. We say that a theta pencil P is δ-nodal if |S| = δ. We say that P is regular if supp(Z) consists of g − 1 distinct points.

If P is a δ-nodal theta pencil, we have an induced map

m′ : P′ ≅ P^{1} → \overline{S}_{g−δ}^{−},

obtained by sending a general C′ ∈ P′ to the moduli point [C′, O_{C′}(\frac{1}{2}h′)] ∈ \overline{S}_{g−δ}^{−}. We note in passing that a theta pencil also induces a map m : P′ → \overline{S}_{g}^{−} defined as follows. Consider the pencil E + P′ having fixed component E. The general member is a quasi-stable curve D ∈ (E + P′) of arithmetic genus g, with exceptional components {E_{i}}_{i=1,...,δ} corresponding to the exceptional divisors of the blow-up ϵ : X′ → X. Then

m(C) := [C ∪ (∪_{i=1}^{δ}E_{i}), η_{E_{i}} = O_{E_{i}}(1), η_{C′} = O_{C′}(\frac{1}{2}h′)] ∈ \overline{S}_{g}^{−}.

These pencils will be used extensively in the proof of Theorem 0.2.

Assume that [X, H] ∈ F_{g} is a general point, in particular Pic(X) = ℤ · H. Then every smooth curve C ∈ |H| is Brill-Noether general, [La], which implies that h^{0}(C, η) = 1, for every odd theta-characteristic η on C. Theta pencils with smooth general member define a locally closed subset in the Grassmannian G(2, H^{0}(S, O_{S}(H)) of lines in |H|. Let Θ^{−}(X, H) be its Zariski closure in G(2, H^{0}(S, O_{S}(H)).

Proposition 3.6. Θ^{−}(X, H) is pure of dimension g − 1.

Proof. Let f : P^{−}(X, H) → |H| be the projection map from the projectivized universal bundle over Θ^{−}(X, H), and V_{g,0}(|H|) ⊂ |H| be the open locus of smooth curves. Under our assumptions f has finite fibres over V_{g,0}(|H|). Thus P^{−}(X, H) has pure dimension g, and Θ^{−}(X, H) has pure dimension g − 1. □

For a general (thus necessarily regular) theta pencil P ∈ Θ^{−}(X, H), we study in more detail the map m : P′ → \overline{S}_{g}^{−}. Let Δ(X, H) ⊂ |H| be the discriminant locus. Since [X, H] ∈ F_{g} is general, Δ(X, H) is an integral hypersurface parameterizing the singular elements of |H|. It is well-known that deg Δ(X, H) = 6g + 18.

Proposition 3.7. Let P ∈ Θ^{−}(X, H) be a general theta pencil with base locus Z. Then every singular curve C ∈ P is nodal. Furthermore,

P · Δ(X, H) = 2(a_{1} + ⋯ + a_{g−1}) + b_{1} + ⋯ + b_{4g+20},

where a_{i} is the parameter point of a curve A_{i} ∈ P having a point of Z as its only singularity, and b_{j} is the parameter point of a curve B_{j} ∈ P such that sing(B_{j}) ⊂ X − Z. Accordingly,

P · α_{0} = 4g + 20 andP · β_{0} = g − 1.

Proof. We set supp(Z) = {p_{1}, …, p_{g−1}}. Since P is regular, for i = 1, …, g − 1, there exists a unique curve A_{i} ∈ P singular at p_{i}. Moreover, for degree reasons, p_{i} is the unique double point of A_{i}. Each pencil T ⊂ |H| having p_{i} in its base locus is a tangent line to Δ(X, H) at A_{i}. Hence the intersection multiplicity (P · Δ(X, H))_{A_{i}} is at least 2. It follows that the assertion to prove is open on any family of pairs (P, [X, H]) such that P ∈ Θ^{−}(X, H). Since F_{g} is irreducible, it suffices to produce one polarized K3 surface (X, H) satisfying this condition.

For this purpose, we use hyperelliptic polarized K3 surfaces (X, H). Consider a rational normal scroll 𝔽 := 𝔽_{a} ⊂ P^{g}, where a ∈ {0, 1} and g = 2n + 1 − a. A general section R ∈ |O_{𝔽}(1)| is a rational normal curve of degree g − 1. From the exact sequence

0 → O_{𝔽}(−2K_{𝔽} − R) → O_{𝔽}(−2K_{𝔽}) → O_{R}(−2K_{𝔽}) → 0,

one finds that there exist a smooth curve B ∈ | − 2K_{𝔽}| and distinct points o_{1}, …, o_{g−1} ∈ B such that the pencil Q ⊂ |O_{𝔽}(R)| of hyperplane sections through o_{1}, …, o_{g−1} cuts out a pencil with simple ramification on B.

Let ρ : X → 𝔽 be the double covering of 𝔽 branched along B. Then X is a K3 surface and |H| := |O_{X}(ρ^{∗}R)| is a hyperelliptic linear system on X of genus g. Then ρ^{∗}(Q) is a regular theta pencil on X with the required properties. □

Since theta pencils cover \overline{S}_{g}^{−} when g ≤ 11 and g ≠ 10, the following consequence of Proposition 3.7 is very suggestive concerning the variation of κ(\overline{S}_{g}^{−}) as g increases, in particular, in highlighting the significance of the case g = 12.

Corollary 3.8. With the same notation as above, we have that P ·K\overline{S}{− \atop g} = 2g −24. In particular general theta pencils of genus g < 12 are K\overline{S}{− \atop g}-negative.

Proof. Use that (P ·λ)\overline{S}{− \atop g} = (π_{∗}(P)·λ)_{\overline{M}_{g}} = g +1, P ·α_{0} = 4g +20 and P ·β_{0} = g −1. □

Proposition 3.9. The locally closed set of nodal theta pencils in Θ^{−}(X, H) is non empty. If P is a general nodal theta pencil, then a general curve C ∈ P has one node as its only singularity.

Proof. We keep the notation from the previous proof and construct a smooth curve B ∈ | − 2K_{𝔽}| and choose general points o, o_{1}, …, o_{g−3} ∈ B, such that the pencil Q ⊂ |O_{𝔽}(R)| of the hyperplane sections through o_{1} + ⋯ + o_{g−3} + 2o cuts out a pencil with simple ramification on B. Then ρ^{∗}(Q) is a nodal theta pencil with the required properties. □

Theorem 3.10. \overline{S}_{g}^{−} is uniruled for g ≤ 11.

Proof. By [M1-4], a general curve [C] ∈ \overline{M}_{g} is embedded in a K3 surface X precisely when g ≤ 9 or g = 11. By Proposition 3.7, C belongs to a theta pencil P ⊂ |O_{X}(C)| (which moreover, is K\overline{S}{− \atop g}-negative). Thus the statement follows for g ≤ 9 and g = 11.

To settle the case of \overline{S}_{10}^{−}, we show that K_{10,1}^{−} is non-empty and irreducible. Indeed, then by Proposition 3.3 it follows that K_{10,1}^{−} is uniruled, and since the projection map K_{10,1}^{−} → S_{10}^{−} is finite, K_{10,1}^{−} dominates S_{10}^{−}. This implies that \overline{S}_{10}^{−} is uniruled.

The variety K_{10,1}^{−} is an open subvariety of the irreducible locus

U := {([C, x, y], η) ∈ M_{10,2} ×_{M_{10}} S_{10}^{−} : h^{0}(C, η ⊗ O_{C}(−x − y)) ≥ 1},

hence it is irreducible as well. To establish its non-emptiness, it suffices to produce an example of an element ([C, x, y], η]) ∈ U, such that the curve C_{xy} can be embedded in a K3 surface. We specialize to the case when C is hyperelliptic and x, y ∈ C are distinct Weierstrass points, in which case one can choose η = O_{C}(x + y + w_{1} + ⋯ + w_{7}), where w_{i} are distinct Weierstrass points in C − {x, y}. Again we let ρ : X → 𝔽 ⊂ P^{11} be a hyperelliptic K3 surface branched along B ∈ | − 2K_{𝔽}|, with polarization H := ρ^{∗}O_{𝔽}(1), so that [X, H] ∈ F_{11}. We set C := ρ^{∗}(R), where R ∈ |O_{𝔽}(1)| is a rational normal curve of degree 10. We need to ensure that C is 1-nodal, with its node p ∈ C such that if f : C′ → C denotes the normalization map, then both points in f^{−1}(p) are Weierstrass points. This is satisfied once we choose R in such a way that B · R ≥ 2ρ(p). □

4. Unirationality of \overline{S}_{g}^{−} for g ≤ 8

To prove the claimed unirationality results, we use that a general curve [C] ∈ \overline{M}_{g} has a sextic plane model when g ≤ 6, or is a linear section of a Mukai variety, when 7 ≤ g ≤ 9. We start with the easy case of small genus, before moving on to the more substantial study of Mukai models.

Theorem 4.1. \overline{S}_{g}^{−} is unirational for g ≤ 6.

Proof. A general odd spin curve [C, η] ∈ \overline{S}_{g}^{−} of genus 3 ≤ g ≤ 6, is birational to a pair (Γ, η), where Γ ⊂ P^{2} is an integral nodal sextic. One can assume that d := supp(η) is a reduced divisor contained in Γ_{reg}. Note that there exists a unique plane cubic E such that E · Γ = 2e, where e is an effective divisor of degree 9 on E, supported on sing(Γ) ∪ d. We denote by U ⊂ (P^{2})^{9} the open set parameterizing general 9-tuples (x̄, ȳ) := (x_{1}, …, x_{δ}, y_{1}, …, y_{g−1}), where g = 10 − δ. Over U lies a projective bundle P whose fibre at (x̄, ȳ) is the linear system of plane sextics Γ which are singular along x̄ and totally tangent to E_{x̄,ȳ} along ȳ. Here E_{x̄,ȳ} ∈ |O_{P}_{2}(3)| denotes the unique plane cubic through the points x_{1}, …, x_{δ}, y_{1}, …, y_{g−1}. Then P is a rational variety, and by the previous remark, it dominates \overline{S}_{g}^{−}. Thus \overline{S}_{g}^{−} is unirational. □

We assume now that 7 ≤ g ≤ 10 and denote by V_{g} ⊂ P^{N_{g}} the rational homogeneous space V_{g} defined as follows [M1], [M2], [M3]:

- V_{10}: the 5-dimensional variety G_{2}/P ⊂ P^{17} corresponding to the Lie group G_{2},

- V_{9}: the Plücker embedding of the symplectic Grassmannian SG(3, 6) ⊂ P^{13},

- V_{8}: the Plücker embedding of the Grassmannian G(2, 6) ⊂ P^{14},

- V_{7}: the Plücker embedding of the orthogonal Grassmannian OG(5, 10) ⊂ P^{15},

Note that N_{g} = g + dim(V_{g}) − 2. Inside the Hilbert scheme Hilb(V_{g}) of curvilinear sections of V_{g}, we consider the open set U_{g} classifying curves C ⊂ V_{g} such that

• C is a nodal integral section of V_{g} by a linear space of dimension g − 1,

• the residue map ρ : H^{0}(C, ω_{C}) → H^{0}(C, ω_{C} ⊗ O_{sing(C)}) is surjective.

A general point [C ↪ P^{g−1}] ∈ U_{g} is a smooth, canonical curve of genus g. Moreover C has general moduli if g ≤ 9. For each 0 ≤ δ ≤ g − 1, we define the locally closed sets of δ-nodal curvilinear sections of V_{g}

U_{g,δ} := {[C ↪ P^{g−1}] ∈ U_{g} : |sing(C)| = δ}.

Proposition 4.2. U_{g,δ} is smooth of pure codimension δ in U_{g}.

Proof. A general 2-dimensional linear section of V_{g} is a polarized K3 surface (S, H) ∈ F_{g} with general moduli. It is known [Ta], that δ-nodal hyperplane sections of S form a pure (g − δ)-dimensional family V_{g,δ}(|H|) ⊂ |H|. In particular U_{g,δ} ≠ ∅ and codim(U_{g,δ}, U_{g}) ≤ δ. We fix a curve [C] ∈ U_{g,δ}, then consider the normal bundle N_{C} of C in V_{g} and the map r : H^{0}(C, N_{C}) → O_{sing(C)} induced by the exact sequence

(3)  0 → T_{C} → T_{V_{g}} ⊗ O_{C} → N_{C} →^{r} T_{C}^{1} → 0,

where T_{C}^{1} = O_{sing(C)} is the Lichtenbaum-Schlessinger sheaf of C. Using the identification T_{[C]}(U_{g}) = H^{0}(C, N_{C}), it is known that Ker(r) is isomorphic to T_{[C]}(U_{g,δ}). We have

that N_{C} ≅ ω_{C}^{⊕(N_{g}−g+1)} and r = ρ^{⊕(N_{g}−g+1)}, where ρ : H^{0}(C, ω_{C}) → H^{0}(C, O_{sing(C)}) is the map given by the residues at the nodes. Since ρ is surjective, Ker(r) has codimension δ inside T_{[C]}(U_{g}) and the statement follows. □

The automorphism group Aut(V_{g}) acts in the natural way on Hilb(V_{g}). Since the locus of singular curvilinear sections [C] ∈ U_{g} is an Aut(V_{g})-invariant divisor which misses a general point of U_{g}, it follows that U_{g}^{ss} := U_{g} ∩ Hilb(V_{g})^{ss} ≠ ∅. Note that since ρ(V_{g}) = 1, the notion of stability is independent of the polarization. The (quasi-projective) GITquotient

𝔐_{g} := U_{g}^{ss}//Aut(V_{g})

is said to be the Mukai model of \overline{M}_{g}. We have the following commutative diagram

U_{g}^{ss} ⟶ U_{g}
u_{g}↓  ^{m_{g}}↓
𝔐_{g} ⟶^{ϕ_{g}} \overline{M}_{g}

where u_{g} : U_{g}^{ss} → 𝔐_{g} is the quotient map and m_{g} : U_{g} → \overline{M}_{g} is the moduli map. The general fibre of m_{g} is an Aut(V_{g})-orbit. Summarizing results from [M1], [M2], [M3], we state the following:

Theorem 4.3. For 7 ≤ g ≤ 9, the map ϕ_{g} : 𝔐_{g} → \overline{M}_{g} is a birational isomorphism. The inverse map ϕ_{g}^{−1} contracts the (unique) Brill-Noether divisor \overline{M}_{g,d}^{r} ⊂ \overline{M}_{g} of curves with a 𝔤_{d}^{r}, as well as the boundary divisors Δ_{i} with 1 ≤ i ≤ [g/2].

Next, let Δ_{g}^{δ} ⊂ Δ_{0} ⊂ \overline{M}_{g} be the locus of integral stable curves of arithmetic genus

g with δ nodes. Then Δ_{g}^{δ} is irreducible of codimension δ in \overline{M}_{g}.

Lemma 4.4. Set g ≤ 9 and let D be any irreducible component of U_{g,δ}. Then the restriction morphism m_{g|D} : D → Δ_{g}^{δ} is dominant. In particular, a general δ-nodal curve [C] ∈ Δ_{g}^{δ} lies on a smooth K3 surface.

Proof. Since U_{g,δ} is smooth, D is a connected component of U_{g,δ}, that is, for [C] ∈ D, the tangent spaces to D and to U_{g,δ} coincide. We consider again the sequence (3):

0 → T_{C} → T_{V_{g}} ⊗ O_{C} → N′_{C} → 0,

where N′_{C} := Im {T_{V_{g}} ⊗ O_{C} → N_{C}} is the equisingular sheaf of C. We have that H^{0}(C, N′_{C}) = Ker(r). As remarked in the proof of Proposition 4.2, H^{0}(C, N′_{C}) is the tangent space T_{[C]}(U_{g,δ}) and its codimension in H^{0}(C, N_{C}) equals δ. Consider the coboundary map ∂ : H^{0}(C, N′_{C}) → H^{1}(C, T_{C}). Since H^{1}(C, T_{C}) classifies topologically trivial deformations of the nodal curve C, the image Im(∂) is isomorphic to the image of the tangent map dm_{g|U_{g,δ}} at [C]. On the other hand H^{0}(C, T_{V_{g}} ⊗ O_{C}) is the tangent space to the orbit of C under the action of Aut(V_{g}). This is reduced and the stabilizer of C, being a subgroup of Aut(C), is finite, hence we obtain:

dim Im(∂) = h^{0}(C, N_{C}) − δ − dim Aut(V_{g}) = 3g − 3 − δ.

Since Δ_{g}^{δ} has codimension δ in \overline{M}_{g}, it follows that m_{g|D} is dominant. □

Proposition 4.5. Fix 0 ≤ δ ≤ g − 1 and D an irreducible component of U_{g,δ}. Then D^{ss} ≠ ∅.

Proof. It suffices to construct an Aut(V_{g})-invariant divisor which does not contain D. We carry out the construction when g = 8, the remaining cases being largely similar.

We fix a complex vector space V ≅ ℂ^{6}, and then V_{8} := G(2, V) ⊂ P(∧^{2}V) and U_{8} ⊂ G(8, ∧^{2}V). For a projective 7-plane Λ ∈ G(8, ∧^{2}V), we denote the set of containing hyperplanes F_{Λ} := {H ∈ P(∧^{2}V)^{∨} : H ⊃ Λ}, and define the Aut(V_{8})-invariant divisor

Z := {Λ ∈ U_{8} : F_{Λ} ∩ G(2, V^{∨}) ⊂ P(∧^{2}V)^{∨} is not a transverse intersection }.

We claim that D ⊈ Z. Indeed, let us fix a general point [C ↪ Λ] ∈ D, where Λ = ⟨C⟩, corresponding to a general curve [C] ∈ Δ_{g}^{δ}. In particular, we may assume that C lies outside the closure in \overline{M}_{g} of curves violating the Petri theorem. Thus C possesses no generalized 𝔤_{7}^{2}’s, that is, \overline{W} _{7}^{2}(C) = ∅, whereas \overline{W} _{5}^{1}(C) ⊂ Pic(C) consists of locally free pencils satisfying the Petri condition. We recall from [M2] the construction of ϕ_{g}^{−1}[C], which generalizes to irreducible Petri general nodal curves: There exists a unique rank two vector bundle E on C with det(E) = ω_{C} and h^{0}(C, E) = 6. This appears as an extension

0 → A → E → ω_{C} ⊗ A^{∨} → 0,

for every A ∈ \overline{W} _{5}^{1}(C). Then one sets ϕ_{g}^{−1}([C]) := [C ↪ G(2, H^{0}(C, E)^{∨})]. Moreover,

F_{Λ} = P(Ker{∧^{2}H^{0}(C, E) → H^{0}(C, ω_{C})}).

In particular, the intersection F_{Λ}∩G(2, H^{0}(C, E)) corresponds to the pencils A ∈ \overline{W} _{5}^{1}(C). Since C is Petri general, \overline{W} _{5}^{1}(C) is a smooth scheme, thus [C ↪ Λ] ∉ Z. □

We consider the quotient 𝔐_{g,δ} := U_{g,δ}^{ss}//Aut(V_{g}) and the induced map
ϕ_{g,δ} : 𝔐_{g,δ} → Δ_{g}^{δ}.

Theorem 4.6. The variety 𝔐_{g,δ} is irreducible and ϕ_{g,δ} is a birational isomorphism.

Proof. By Lemma 4.4, any irreducible component Y of 𝔐_{g,δ} dominates Δ_{g}^{δ}. On the other hand, ϕ_{g} : 𝔐_{g} → \overline{M}_{g} is a birational morphism and ϕ_{g,δ} = ϕ_{g|𝔐_{g,δ}}. Since \overline{M}_{g} is normal, each fibre of ϕ_{g} is connected, thus 𝔐_{g,δ} is irreducible and deg(ϕ_{g,δ}) = 1. □

We lift our construction to the space of odd spin curves. Keeping g ≤ 9, we consider the Hilbert scheme Hilb_{2g−2}(V_{g}) of 0-dimensional subschemes of V_{g} having length 2g − 2.

Definition 4.7. Let ℨ_{g−1} ⊂ Hilb_{2g−2}(V_{g}) be the parameter space of those 0-dimensional schemes Z ⊂ V_{g} such that:

(1) Z is a hyperplane section of a smooth curve section [C] ∈ U_{g},

(2) Z has multiplicity two at each point of its support,

(3) supp(Z) consists of g − 1 linearly independent points.

One thinks of ℨ_{g−1} as classifying length g − 1 clusters on V_{g}. A general point of ℨ_{g−1} corresponds to a 0-cycle x_{1} + ⋯ + x_{g−1} ∈ Sym^{g−1}(V_{g}) satisfying

dim ⟨x_{1}, …, x_{g−1}⟩ ∩ 𝕋_{x_{i}}(V_{g}) ≥ 1, for i = 1, …, g − 1.

Clearly dim(ℨ_{g−1}) = (g−1)(N_{g}−g+1). Then we consider the incidence correspondence

U_{g}^{−} := {(C, Z) ∈ U_{g} × ℨ_{g−1} : Z ⊂ C}.

The first projection map π_{1} : U_{g}^{−} → U_{g} is finite of degree 2^{g−1}(2^{g}−1); its fibre at a general point [C] ∈ U_{g} is in bijective correspondence with the set of odd theta-characteristics of C. In particular, both U_{g}^{−} and ℨ_{g−1} are irreducible varieties. The spin moduli map

m_{g}^{−} : U_{g}^{−} −−→ \overline{S}_{g}^{−}

is defined by m_{g}^{−}(C, Z) := [C, O_{C}(Z/2)], for each point (C, Z) ∈ U_{g}^{−} corresponding to a smooth curve C. Later we shall extend the rational map m_{g}^{−} to a regular map over U_{g}^{−}. It is clear that m_{g}^{−} induces a map ϕ_{g}^{−} : Q_{g}^{−} −−→ \overline{S}_{g}^{−} from the quotient

Q_{g}^{−} := π_{1}^{−1}(U_{g}^{ss})//Aut(V_{g}).

We may think of Q_{g}^{−} as being the Mukai model of \overline{S}_{g}^{−}. If π^{−} : Q_{g}^{−} → 𝔐_{g} is the map induced by π at the level of Mukai models, we have a commutative diagram:

Q_{g}^{−} ⟶^{ϕ_{g}^{−}} \overline{S}_{g}^{−}
π^{−}↓  ^{π}↓
𝔐_{g} ⟶^{ϕ_{g}} \overline{M}_{g}

Proposition 4.8. The spin Mukai model Q_{g}^{−} is irreducible and ϕ_{g}^{−} : Q_{g}^{−} → \overline{S}_{g}^{−} is a birational isomorphism.

One extends the rational map m_{g}^{−} (therefore ϕ_{g}^{−} as well) to a regular morphism as follows. Let (C, Z) ∈ U_{g}^{−} be an arbitrary point, and set supp(Z) := {p_{1}, …, p_{g−1}}. Assume that sing(C) ∩ supp(Z) = {p_{1}, …, p_{δ}}, where δ ≤ g − 1. Consider the partial normalization ν : N → C at the points p_{1}, …, p_{δ}. In particular, there exists an effective Cartier divisor e on C of degree g − δ − 1, such that 2e = Z ∩ (C − sing(C)), and set ϵ := O_{N}(ν^{∗}e). Then m_{g}^{−}(C, Z) is the spin curve [X, η] ∈ \overline{S}_{g}^{−} defined as follows:

Definition 4.9.

(1) X := N ∪ E_{1} ∪ ⋯ ∪ E_{δ}, where E_{i} = P^{1} for i = 1, …, δ.

(2) E_{i} ∩ N = ν^{−1}(p_{i}), for every node p_{i} ∈ sing(C) ∩ supp(Z).
(3) η ⊗ O_{N} ≅ ϵ and η ⊗ O_{E_{i}} ≅ O_{P^{1}}(1).

We note that N is smooth of genus g − δ, precisely when sing(C) ⊂ supp(Z). In this case ϵ ∈ Pic^{g−1−δ}(N) is a theta characteristic and h^{0}(N, ϵ) = 1. Since we are specially interested in this case, for 1 ≤ δ ≤ g − 1 we introduce the locally closed sets

U_{g,δ}^{−} := {(C, Z) ∈ U_{g}^{−} : sing(C) ⊂ supp(Z), |sing(C)| = δ}.

We denote by B_{g,δ}^{−} the closure of m_{g}^{−}(U_{g,δ}^{−}) inside \overline{S}_{g}^{−}; this is the closure in \overline{S}_{g}^{−} of the locus of δ-nodal spin curves having δ exceptional components. Clearly B_{g,δ}^{−} is an irreducible component of π^{−1}(Δ_{g}^{δ}). We set

Q_{g,δ}^{−} := U_{g,δ}^{−} ∩ π_{1}^{−1}(U_{g}^{ss})//Aut(V_{g}),

and let u_{g}^{−} : U_{g,δ}^{−} −−→ Q_{g,δ}^{−} denote the quotient map. Keeping all previous notation, we have a further commutative diagram

U_{g,δ}^{−} ⟶^{u_{g}^{−}} Q_{g,δ}^{−} ⟶^{ϕ_{g,δ}^{−}} B_{g,δ}^{−}
↓  _{π^{−}}↓  ^{π}↓
U_{g,δ} ⟶^{u_{g}} 𝔐_{g,δ} ⟶^{ϕ_{g,δ}} Δ_{g}^{δ}

where ϕ_{g,δ}^{−} is the morphism induced on Q_{g,δ}^{−} by m_{g}^{−}.

Theorem 4.10. We fix 7 ≤ g ≤ 9 and 1 ≤ δ ≤ g − 1. Then ϕ_{g,δ}^{−} : Q_{g,δ}^{−} → B_{g,δ}^{−} is a birational isomorphism.

Proof. It suffices to note that ϕ_{g,δ} is birational, and the vertical arrows of the diagram are finite morphisms of the same degree, namely the number of odd theta-characteristics on a curve of genus g − δ. □

We construct a projective bundle over B_{g,δ}^{−}, then show that for certain values δ ≤ g − 1, the locus B_{g,δ}^{−} itself is unirational, whereas the above mentioned bundle dominates S_{g}^{−}. Let C_{g,δ} ⊂ U_{g,δ}^{−} × V_{g} be the universal curve, endowed with its two projection maps

U_{g,δ}^{−} ⟵^{p} C_{g,δ} ⟶^{q} V_{g}.

We fix an arbitrary point (Γ, Z) ∈ U_{g,δ}^{−} and let ν : N → Γ be the normalization map. Recall that sing(Γ) consists of δ linearly independent points and that h^{0}(N, O_{N}(ν^{∗}e)) = 1, where e is the effective divisor on Γ characterized by Z_{|Γ_{reg}} = 2e. Thus the restriction map H^{0}(Γ, ω_{Γ}) → H^{0}(ω_{Γ} ⊗ O_{Z}) has 1-dimensional kernel. In particular the relative cotangent sheaf ω_{p} admits a global section s inducing an exact sequence

0 → O_{C_{g,δ}} → ω_{p} → O_{W} ⊗ ω_{p} → 0,

which defines a subscheme W ⊂ C_{g,δ}, whose fibre at the point (Γ, Z) ∈ U_{g,δ}^{−} is Z itself. We set

A := p_{∗}(I_{W/C_{g,δ}} ⊗ q^{∗}O_{V_{g}}(1)),

which is a vector bundle on U_{g,δ}^{−} of rank N_{g} − g + 2. The fibre of A(Γ, Z) is identified with H^{0}(V_{g}, I_{Z/V_{g}}(1)). One has a natural identification

PH^{0}(I_{Z/V_{g}}(1))^{∨} = {1-dimensional linear sections of V_{g} containing Z }.

Definition 4.11. P_{g,δ} is the projectivized dual of A.

From the definitions and the previous remark it follows:

Proposition 4.12. P_{g,δ} is the Zariski closure of the incidence correspondence

P_{g,δ}^{o} := {(C, (Γ, Z)) ∈ U_{g} × U_{g,δ}^{−} : Z ⊂ C}.

Consider the projection maps

U_{g}^{−} ⟵^{α} P_{g,δ}^{o} ⟶^{β} U_{g,δ}^{−}.

We wish to know when α is a dominant map. For 1 ≤ δ < g ≤ 9, we have the following:

Proposition 4.13. The map α is dominant if and only if δ ≤ N_{g} + 1 − g = dim(V_{g}) − 1.

Proof. By definition, the morphism β is surjective. Let (Γ, Z) ∈ U_{g,δ}^{−} be an arbitrary point, and set sing(Γ) := {p_{1}, …, p_{δ}} ⊂ Z. We define P_{Z} to be the locus of 1-dimensional linear sections of V_{g} containing Z. Inside P_{Z} we consider the space

P_{Γ,Z} = {Γ′ ∈ P_{Z} : sing(Γ′) ∩ Z ⊇ sing(Γ) ∩ Z},

First note that for p ∈ sing(Γ), the locus H_{p} := {Γ′ ∈ P_{Z} : p ∈ sing(Γ′)} is a hyperplane in P_{Z}. Indeed, we identify P_{Z} with the family of linear spaces L ∈ G(g, N_{g} + 1) such that ⟨Z⟩ ⊂ L. By the definition of the cluster Z, it follows that l := 𝕋_{p}(V_{g}) ∩ ⟨Z⟩ is a line. For L ∈ P_{Z}, the intersection L ∩ V_{g} is singular at p if and only if dim L ∩ 𝕋_{p}(V_{g}) ≥ 2. This is obviously a codimension 1 condition in P_{Z}. Therefore, if for 1 ≤ i ≤ δ we define the hyperplane H_{i} := {L = ⟨Γ′⟩ ∈ P_{Z} : dim L ∩ 𝕋_{p_{i}}(V_{g}) ≥ 2}, then

P_{Γ,Z} = H_{1} ∩ ⋯ ∩ H_{δ}.

This shows that the general point in β^{−1}(C, Z) corresponds to a smooth curve C ⊃ Z. We now fix a general point (Γ, Z) ∈ U_{g,δ}^{−}, corresponding to a general cluster Z ∈ ℨ_{g−1}.

Claim: P_{Γ,Z} has codimension δ in P_{Z}; its general element is a nodal curve with δ nodes. Proof of claim: Indeed P_{Z} is a general fibre of the projective bundle U_{g}^{−} → ℨ_{g−1}. The claim follows since codim(U_{g,δ}^{−}, U_{g}^{−}) = δ.

The fibre α^{−1}((C, Z)) over a general point (C, Z) ∈ U_{g}^{−}, is the union of \binom{g−1}{δ} linear spaces H_{1} ∩ ⋯ ∩ H_{δ} ⊂ P_{Z} as above. By the claim above, when Z ∈ ℨ_{g−1} is a general cluster, this is a union of linear spaces P_{Γ,Z} as before, having codimension δ in P_{Z}. Hence α^{−1}((C, Z)) is not empty if and only if δ ≤ dim P_{Z}, that is, δ ≤ N_{g} − g + 1. □

Let us fix the following notation:

Definition 4.14.

(1) \overline{ℙ}_{g,δ} := (P_{g,δ}^{o})^{ss}//Aut(V_{g}).

(2) \overline{β} : \overline{ℙ}_{g,δ} → \overline{S}_{g}^{−} is the morphism induced by β at the level of quotients.

Note that β : P_{g,δ} → U_{g,δ}^{−} is a projective bundle and Aut(V_{g}) acts linearly on its fibres, therefore β descends to a projective bundle on B_{g,δ}^{−}. Then it follows from the previous remark that P_{g,δ} is birationally isomorphic to P^{N_{g}−g+1} × B_{g,δ}^{−}. To finish the proof of the unirationality of S_{g}^{−}, we proceed as follows:

Theorem 4.15. Let 7 ≤ g ≤ 9 and assume that (i) B_{g,δ}^{−} is unirational and (ii) δ ≤ N_{g} − g + 1.

Then \overline{S}_{g}^{−} is unirational.

Proof. By assumption (ii), β : P_{g,δ}^{o} → U_{g}^{−} is dominant, Hence the same is true for the

induced morphism \overline{β} : \overline{ℙ}_{g,δ} → \overline{S}_{g}^{−}. By (i) and the above remark, \overline{ℙ}_{g,δ} is unirational. Therefore \overline{S}_{g}^{−} is unirational as well. □

Theorem 4.15 has some straightforward applications. The case δ = g − 1 is particularly convenient, since B_{g,g−1}^{−} is isomorphic to the moduli space of integral curves of geometric genus 1 with g − 1 nodes. For δ = g − 1, the assumptions of Theorem 4.15 hold when g ≤ 8. In this range, the unirationality of S_{g}^{−} follows from that of B_{g,g−1}^{−}.

Theorem 4.16. B_{g,g−1}^{−} is unirational for g ≤ 10.

Proof. Let I ⊂ P^{2} × (P^{2})^{∨} be the natural incidence correspondence consisting of pairs (x, l) such that x is a point on the line l. For δ ≤ 9, we define

Π_{δ} := {(x_{1}, l_{1}, …, x_{δ}, l_{δ}, E) ∈ I^{δ} × PH^{0}(P^{2}, O_{P}_{2}(3)) : x_{1}, …, x_{δ} ∈ E}.

Then there exists a rational map f_{δ} : Π_{δ} −−→ B_{δ+1,δ}^{−} sending (x_{1}, l_{1}, …, x_{δ}, l_{δ}, E) to the moduli point of the δ-nodal, integral curve C obtained from the elliptic curve E, by identifying the pairs of points in E ∩ l_{i} − {x_{i}} for 1 ≤ i ≤ δ. It is easy to see that Π_{δ} is rational if δ ≤ 9. Clearly f_{δ} is dominant, just because every elliptic curve can be realized as a plane cubic. It follows that B_{δ+1,δ}^{−} is unirational when δ ≤ 9. □

Unfortunately one cannot apply Theorem 4.16 to the case g = 9, since the assumptions of Theorem 4.15 are satisfied only if δ ≤ 5.

5. The stack of degenerate odd theta-characteristics

In this section we define a Deligne-Mumford stack X_{g} → \overline{S}_{g}^{−} parameterizing limit linear series 𝔤_{g−1}^{0} which appear as limits of degenerate theta-characteristics on smooth curves. The push-forward of [X_{g}] is going to be precisely our divisor \overline{Z}_{g}. Having a good description of X_{g} over the boundary will enable us to determine all the coefficients in the expression of [\overline{Z}_{g}] in Pic(\overline{S}_{g}^{−}).

We first define a partial compactification M̃_{g} := M_{g} ∪ Δ̃_{0} ∪ … ∪ Δ̃_{[g/2]} of \overline{M}_{g}, obtaining by adding to M_{g} the open sub-stack Δ̃_{0} ⊂ Δ_{0} of one-nodal irreducible curves [C_{yq} := C/y ∼ q], where [C, y, q] ∈ M_{g−1,2} is a Brill-Noether general curve together with their degenerations [C ∪ D_{∞}] where D_{∞} is an elliptic curve with j(D_{∞}) = ∞, as well as the open substacks Δ̃_{j} ⊂ Δ_{j} for 1 ≤ j ≤ [g/2] classifying curves [C ∪_{y} D] where [C] ∈ M_{j} and [D] ∈ M_{g−j} are Brill-Noether general curves in the respective moduli spaces. Let p : M̃_{g,1} → M̃_{g} be the universal curve. We denote S̃_{g}^{−} := π^{−1}(M̃_{g}) ⊂ \overline{S}_{g}^{−} and note that for all 0 ≤ j ≤ [g/2] the boundary divisors A′_{j} := A_{j} ∩ S̃_{g}^{−}, B′_{j} := B_{j} ∩ S̃_{g}^{−}

are mutually disjoint inside S̃_{g}^{−}. Finally, we consider Z := S̃_{g}^{−} ×_{M̃_{g}} M̃_{g,1} and denote by

p_{1} : Z → S̃_{g}^{−} the projection.

Following the local description of the projection \overline{S}_{g}^{−} → \overline{M}_{g} carried out in [C], in

order to obtain the universal spin curve over S̃_{g}^{−} one has first to blow-up the codimension 2 locus V ⊂ Z corresponding to points

v = ([C∪_{{y,q}}E, η_{C}^{⊗2} = K_{C}, h^{0}(η_{C}) ≡ 1 mod 2, η_{E} = O_{E}(1)], ν(y) = ν(q)) ∈ B′_{0}×_{M̃_{g}} M̃_{g,1}

(recall that ν : C → C_{yq} denotes the normalization map, so v corresponds to the marked point specializing to the node of the curve C_{yq}). Suppose that (τ_{1}, …, τ_{3g−3}) are local coordinates in an étale neighbourhood of [C ∪_{{y,q}} E, η_{C}, η_{E}] ∈ S̃_{g}^{−} such that the local equation of the divisor B′_{0} is (τ_{1} = 0). Then Z around v admits local coordinates (x, y, τ_{1}, …, τ_{3g−3}) verifying the equation xy = τ_{1}^{2}, in particular, Z is singular along V. Next, for 1 ≤ j ≤ [g/2] one blows-up the codimension 2 loci V_{j} ⊂ Z consisting of points

([C ∪_{q} D, η_{C}, η_{D}], q ∈ C ∩ D) ∈ (A′_{j} ∪ B′_{j}) ×_{M̃_{g}} M̃_{g,1}.

This corresponds to inserting an exceptional component in each spin curve in π^{∗}(Δ̃_{j}). We denote by

C := Bl_{V ∪V_{1}∪...∪V_{[g/2]}}(Z)

and by f : C → S̃_{g}^{−} the induced family of spin curves. Then for every [X, η, β] ∈ S̃_{g}^{−} we have an isomorphism between f^{−1}([X, η, β]) and the quasi-stable curve X.

There exists a spin line bundle P ∈ Pic(C) of relative degree g − 1 as well as a morphism of O_{C}-modules B : P^{⊗2} → ω_{f} having the property that P_{|f^{−1}([X,η,β])} = η and B_{|f^{−1}([X,η,β])} = β : η^{⊗2} → ω_{X}, for all spin curves [X, η, β] ∈ S̃_{g}^{−}. We note that for the even moduli space S̃_{g}^{+} one has an analogous construction of the universal spin curve.

Next we define the stack τ : X_{g} → S̃_{g}^{−} classifying limit 𝔤_{g−1}^{0} which are twists of degenerate odd-spin curves. For a tree-like curve X we denote by \overline{G}_{d}^{r}(X) the scheme of limit linear series 𝔤_{d}^{r}. The fibres of the morphism τ have the following description:

• τ^{−1}(S_{g}^{−}) parameterizes triples ([C, η], σ, x), where [C, η] ∈ S_{g}^{−}, x ∈ C is a point and σ ∈ PH^{0}(C, η) is a section such that div(σ) ≥ 2x.

• For 1 ≤ j ≤ [g/2] the inverse image τ^{−1}(A′_{j} ∪ B′_{j}) parameterizes elements of the form

(X, σ ∈ \overline{G}_{g−1}^{0}(X), x ∈ X_{reg}),

where (X, x) is a 1-pointed quasi-stable curve semistably equivalent to the underlying curve of a spin curve [C ∪_{q} E ∪_{q′} D, η_{C}, η_{E}, η_{D}] ∈ A′_{j} ∪ B′_{j}, with E denoting the exceptional component, g(C) = j, g(D) = g − j, {q} = C ∩ E, {q′} = E ∩ D and

σ_{C} ∈ PH^{0}(C, η_{C} ⊗O_{C}((g −j)q)), σ_{D} ∈ PH^{0}(D, η_{D} ⊗O_{D}(jq′)), σ_{E} ∈ PH^{0}(E, O_{E}(g −1))

are aspects of the limit linear series σ on X. Moreover, we require that ord_{x}(σ) ≥ 2.

• τ^{−1}(B′_{0}) parameterizes elements (X, η ∈ Pic^{g−1}(X), σ ∈ PH^{0}(X, η), x ∈ X_{reg}), where (X, x) is a 1-pointed quasi-stable curve equivalent to the curve underlying a point [C ∪_{{y,q}} E, η_{C}, η_{E}] ∈ B′_{0}, the line bundle η on X satisfies η_{|C} = η_{C} and η_{|E} = η_{E} and η_{|Z} = O_{Z} for the remaining components of X. Finally, we require ord_{x}(σ) ≥ 2.

• τ^{−1}(A′_{0}) corresponds to points (X, η ∈ Pic^{g−1}(X), σ ∈ PH^{0}(X, η), x ∈ X_{reg}), where (X, x) is a 1-pointed quasi-stable curve equivalent to the curve underlying a point [C_{yq}, η_{C_{yq}}] ∈ A′_{0}, and if μ : X → C_{yq} is the map contracting all exceptional components, then μ^{∗}(η_{C_{yq}}) = η (in particular η is trivial along exceptional components), and finally ord_{x}(σ) ≥ 2.

Using general constructions of stacks of limit linear series cf. [EH1], [F2], it is clear that X_{g} is a Deligne-Mumford stack. There exists a proper morphism

τ : X_{g} → S̃_{g}^{−}

that factors through the universal curve and we denote by χ : X_{g} → C the induced morphism, hence τ = f ∘ χ. The push-forward of the coarse moduli space τ_{∗}([X_{g}]) equals scheme-theoretically \overline{Z}_{g} ∩ S̃_{g}^{−}. It appears possible to extend X_{g} over the entire \overline{S}_{g}^{−} but this is not necessary in order to prove Theorem 0.3 and we skip the details.

We are now in a position to calculate the class of the divisor \overline{Z}_{g} and we expand its class in the Picard group of \overline{S}_{g}^{−}

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

where λ̄, ᾱ_{i}, β̄_{i} ∈ ℚ for i = 0, …, [g/2]. We start by determining the coefficients of the divisors α_{i} and β_{i} for 1 ≤ i ≤ [g/2].

Proposition 5.1. For 1 ≤ i ≤ [g/2] we have that F_{i} · \overline{Z}_{g} = 4(g − i)(i − 1) and the intersection is everywhere transverse. It follows that ᾱ_{i} = 2(g − i).

Proof. We recall from the definition of F_{i} that we have fixed theta-characteristics of opposite parity η_{C}^{−} ∈ Pic^{i−1}(C) and η_{D}^{+} ∈ Pic^{g−i−1}(D). We choose a point t = (X, η, σ, x) ∈ τ^{−1}(F_{i}). It is a simple exercise to show that the ”double” point x of σ ∈ \overline{G}_{g−1}^{0}(X) cannot specialize to the exceptional component, therefore one has only two cases to consider depending on whether x lies on C or on D. Assume first that x ∈ C and then σ_{C} ∈ PH^{0}(C, η_{C}^{−} ⊗ O_{C}((g − i)q)) and σ_{D} ∈ PH^{0}(D, η_{D}^{+} ⊗ O_{D}(iq)), where {q} = C ∩ D is a point which moves on C but is fixed on D. Then ord_{q}(σ_{D}) ≤ i − 1, therefore ord_{q}(σ_{C}) ≥ g − i and then σ_{C}(−(g − i)q) ∈ PH^{0}(C, η_{C}^{−}). In particular, if we choose [C, η_{C}^{−}] ∈ S_{i} − Z_{i}, then the section σ_{C}(−(g − i)q) has only simple zeros, which shows that x cannot lie on C, so this case does not occur.

We are left with the possibility x ∈ D − {q}. One quickly concludes that the only possibility is ord_{q}(σ_{C}) = g − i + 1 and ord_{q}(σ_{D}) = i − 2. In particular, q ∈ supp(η_{C}^{−}) which gives i − 1 choices for the moving point q ∈ C. Furthermore σ_{D}(−(i − 2)q) ∈ H^{0}(D, η_{D}^{+} ⊗ O_{D}(2q − 2x)), that is, x specializes to one of the ramification points of the pencil η_{D}^{+} ⊗ O_{D}(2q) ∈ W_{g−i+1}^{1}(D). We note that because of the generality of [D, η_{D}^{+}] ∈ S_{g−i}^{+} as well as that of q ∈ D, the pencil is base point free and complete. From the Hurwitz-Zeuthen formula one finds 4(g −i) ramification points of |η_{D}^{+} ⊗O_{D}(2q)|, which leads to the formula F_{i} · \overline{Z}_{g} = 4(g − i)(i − 1). The fact that τ_{∗}(X_{g}) is transverse to F_{i} follows because the formation of X_{g} commutes with restriction to B′_{0} and then one can easily show in a way similar to [EH2] Lemma 3.4 or by direct calculation that X_{g} ×S̃{− \atop g} B′_{0}

is smooth at any of the points in τ^{−1}(F_{i}). □

Proposition 5.2. For 1 ≤ i ≤ [g/2] we have that G_{i} · \overline{Z}_{g} = 4i(i − 1) and the intersection is transversal. In particular β̄_{i} = 2i.

Proof. This time we fix general points [C, η_{C}^{+}] ∈ S_{i}^{+} and [D, η_{D}^{−}] ∈ S_{g−i}^{−} and q ∈ C ∩ D which is a fixed general point on D but an arbitrary point on C. Again, it is easy to see that if t = (X, σ, x) ∈ τ^{−1}(G_{i}) then x must lie either on C or on D. Assume first that x ∈ C − {q}. Then the aspects of σ are described as follows

σ_{C} ∈ PH^{0}(C, η_{C}^{+} ⊗ O_{C}((g − i)q)), σ_{D} ∈ PH^{0}(D, η_{D}^{−} ⊗ O_{D}(iq))

and moreover ord_{x}(σ_{C}) ≥ 2. The point q ∈ D can be chosen so that it does not lie in supp(η_{D}^{−}), hence ord_{q}(σ_{D}) ≤ i and then ord_{q}(σ_{C}) ≥ g−i−1. This leads to the conclusion H^{0}(C, η_{C}^{+} ⊗O_{C}(y −2x)) ≠ 0, or equivalently (x, y) ∈ C ×C is a ramification point of the degree i covering p_{1} : T_{η_{C}^{+}} → C from the associated Scorza curve. We have shown that T_{η_{C}^{+}} is smooth of genus 1 + 3i(i − 1) (cf. Theorem 2.1) and moreover all the ramification points of p_{1} are ordinary, therefore we find

deg Ram(p_{1}) = 2g(T_{C_{η{+ \atop C}}}) − 2 − deg(p_{1})(2i − 2) = 4i(i − 1)

choices when x ∈ C. Next possibility is x ∈ D − {q}. The same reasoning as above shows that ord_{q}(σ_{C}) ≤ g − i − 1, therefore ord_{q}(σ_{D}) ≥ i as well as ord_{x}(σ_{D}) ≥ 2. Since σ_{D}(−iq) ∈ PH^{0}(D, η_{D}^{−}), this case does not occur if [D, η_{D}^{−}] ∈ S_{g−i}^{−} − Z_{g−i}. □

Next we prove that \overline{Z}_{g} is disjoint from both elliptic pencils F_{0} and G_{0}:

Proposition 5.3. We have that F_{0} ·\overline{Z}_{g} = 0 and G_{0} ·\overline{Z}_{g} = 0. The equalities ᾱ−12ᾱ_{0} +ᾱ_{1} = 0 and 3ᾱ − 12ᾱ_{0} − 12β̄_{0} + 3β̄_{1} = 0 follow.

Proof. We first show that F_{0} ∩ \overline{Z}_{g} = ∅ and we assume by contradiction that there exists t = (X, σ, x) ∈ τ^{−1}(F_{0}). Let us deal first with the case when st(X) = C ∩ E_{λ}, with E_{λ} being a smooth curve of genus 1. The key point is that the point of attachment q ∈ C ∩ E_{λ} being general, we can assume that (x, q) ∉ Ram{p_{1} : T_{η_{C}^{+}} → C}, for all x ∈ C. This implies that H^{0}(C, η_{C}^{+} ⊗ O_{C}(q − 2x)) = 0 for all x ∈ C, therefore a section σ_{C} ∈ PH^{0}(C, η_{C}^{+} ⊗ O_{C}(q)) cannot vanish twice anywhere. Thus either x ∈ E_{λ} − {q} or x lies on some exceptional component of X. In the former case, since ord_{q}(σ_{C}) = 0, it follows that ord_{q}(σ_{E_{λ}}) ≥ g − 1, that is, σ_{E_{λ}} has no zeroes other than q (simple or otherwise). In the latter case, when x ∈ E, with E being an exceptional component, we denote by q′ ∈ E the point of intersection of E with the connected subcurve of X containing C as a subcomponent. Since as above, ord_{q}(σ_{C}) = 0, by compatibility it follows that ord_{q′}(σ_{E}) = g − 1. But σ_{E} ∈ PH^{0}(E, O_{E}(g − 1)), that is, σ_{E} does not vanish at x, a contradiction. The proof that G_{0} ∩ \overline{Z}_{g} = ∅ is similar and we omit the details. □

The trickiest part in the calculation of [\overline{Z}_{f}] is the computation of the following intersection number:

Proposition 5.4. If H_{0} ⊂ B_{0} is the covering family lying in the ramification divisor of \overline{S}_{g}^{−}, then one has that H_{0} · \overline{Z}_{g} = 2(g − 2) and the intersection consists of g − 2 points each counted with multiplicity 2. Therefore the relation (g − 1)β̄_{0} − β̄_{1} = 2(g − 2) holds.

Proof. We first determine the set-theoretic intersection τ_{∗}(X_{g}) ∩ H_{0}. We recall that we have fixed [C, q, η_{C}^{−}] ∈ S_{g−1,1}^{−} and start by choosing an arbitrary point t = (X, η, σ, x) ∈ τ^{−1}(H_{0}). Assume first that X = C ∪_{{y,q}} E, where y ∈ C, that is, x does not specialize to one of the nodes of C ∪ E. Suppose first that x ∈ C − {y, q}. From the Mayer-Vietoris sequence on X we write

0 ≠ σ ∈ H^{0}(X, η⊗O_{X}(−2x)) = Ker{H^{0}(C, η_{C}^{−}⊗O_{C}(−2x))⊕H^{0}(E, O_{E}(1)) ⟶^{ev_{y,q}} ℂ_{{y,q}}^{2}},

we obtain that H^{0}(C, η_{C}^{−} ⊗ O_{C}(−2x)) ≠ 0. This case can be avoided by choosing [C, η_{C}^{−}] ∈ S_{g−1}^{−} − Z_{g−1}.

Next we consider the possibility x ∈ E − {y, q}. The same Mayer-Vietoris argument reads in this case 0 ≠ Ker{H^{0}(C, η_{C}^{−}) ⊕ H^{0}(E, O_{E}(−1)) ⟶^{ev_{y,q}} ℂ_{{y,q}}^{2}}, that is, y + q ∈ supp(η_{C}^{−}). This case can be avoided as well by starting with a general point q ∈ C − supp(η_{C}^{−}). Thus the only possibility is that x specializes to one of the nodes y or q.

We deal first with the case when x and q coalesce and there is no loss of generality in assuming that X = C ∪ E ∪ E′, where both E and E′ are copies of P^{1} and C ∩ E = {y}, C ∩ E′ = {q}, E ∩ E′ = {y′} and moreover x ∈ E′ − {y′, q}. The restrictions of the line bundle η ∈ Pic^{g−1}(X) are such that η_{|C} = η_{C}^{−}, η_{E} = O_{E}(1) and η_{E′} = O_{E′}. We write

0 ≠ σ = (σ_{C}, σ_{E}, σ_{E′}) ∈ Ker{H^{0}(C, η_{C}^{−})⊕H^{0}(E, O_{E}(1))⊕H^{0}(E′, O_{E′}(1)) ⟶^{ev_{y,y′,q}} ℂ_{y,y′,q}},

hence σ_{E′} = 0, and then by compatibility σ_{C}(q) = 0, that is, q ∈ supp(η_{C}^{−}) and again this case can be ruled out by a suitable choice of q. The last possible situation is when x and the moving point y ∈ C coalesce, in which case X = C ∪ E ∪ E′, where this time C ∩E = {q}, C ∩E′ = {y}, E ∩E′ = {y′} and again x ∈ E′−{y′, q}. Writing one last time the Mayer-Vietoris sequence we find that σ_{E′} = 0 and then σ_{E}(y′) = 0 and σ_{C}(y) = 0, that is, y ∈ supp(η_{C}^{−}) and then σ_{C} is uniquely determined up to a constant. Finally σ_{E} ∈ H^{0}(E, O_{E}(1)(−y′)) is also uniquely specified by the gluing condition σ_{E}(q) = σ_{C}(q). All in all, H_{0} ∩ \overline{Z}_{g} = #supp(η_{C}^{−}) = g − 2.

This discussion singles out an irreducible component Ξ ⊂ χ_{∗}(X_{g}) ⊂ C of the intersection χ(X_{g}) ∩ f^{−1}(B′_{0}), namely

Ξ = {([C ∪_{{y,q}} E, η_{C}, η_{E}], x) : y ∈ supp(η_{C}^{−}), x = y ∈ X_{sing}},

where we recall that f : C → S̃_{g}^{−} is the universal spin curve. Since Ξ ⊂ Sing(χ_{∗}(X_{g})), it follows after a simple local analysis that each point in τ^{−1}(H_{0}) should be counted with multiplicity 2. □

Remark 5.5. An partial independent check of Theorem 0.4 is obtained by computing using the Porteous formula the coefficient λ̄ in the expression of [\overline{Z}_{g}]. By an abuse of notation we still denote by f : C → S_{g}^{−} the restriction of the universal spin curve to the locus of smooth curves and η ∈ Pic(C) the spin bundle of relative degree g − 1. Then Z_{g} is the push-forward via f : C → S_{g}^{−} of the degeneration locus of the sheaf morphism ϕ : f_{∗}(η) → J_{1}(η) (both these sheaves are locally free away a subset of codimension 3 in S_{g}^{−} and throwing away this locus has no influence on divisor class calculations). Since det(f_{∗}η) = (f_{∗}η)^{⊗2}, it follows that c_{1}(f_{∗}(η)) = −λ/4, whereas the Chern classes of the first jet bundle J_{1}(η) are calculated using the standard exact sequence on C

0 ⟶ η ⊗ ω_{f} ⟶ J_{1}(η) ⟶ η ⟶ 0.

Remembering Mumford’s formula f_{∗}(c_{1}^{2}(ω_{f})) = 12λ, one finally writes that

[Z_{g}] = f_{∗}c_{2}(J_{1}(η) − f_{∗}(η)) = f_{∗}(\frac{3}{4}c_{1}(ω_{f})^{2} − 2c_{1}(ω_{f}) · c_{1}(f_{∗}(η))) = (g + 8) λ ∈ Pic(S_{g}^{−}).

6. A divisor of small slope on \overline{M}_{12}

The aim of this section is to construct an effective divisor D ∈ Eff(\overline{M}_{12}) of slope s(D) < 6 + 12/13, that is, violating the Slope Conjecture. As pointed out in the proof of Theorem 0.3, this is precisely what is needed to show that \overline{S}_{12}^{−} is of general type.

Theorem 6.1. The following locus consisting of curves of genus 12

𝔇_{12} := {[C] ∈ M_{12} : ∃L ∈ W_{14}^{4}(C) withμ_{0}(L) : Sym^{2}H^{0}(C, L) → H^{0}(C, L^{⊗2}) not injective}

is a divisor on M_{12}. The class of its compactification inside \overline{M}_{12} equals

\overline{𝔇}_{12} ≡ 13245 λ − 1926 δ_{0} − 9867 δ_{1} − ∑_{j=2}^{6} b_{j} δ_{j} ∈ Pic(\overline{M}_{12}),

where b_{j} ≥ b_{1} for j ≥ 2. In particular, s(\overline{𝔇}_{12}) = \frac{4415}{642} < 6 + \frac{12}{13}.

This implies the following upper bound for the slope s(\overline{M}_{12}) of the moduli space:

Corollary 6.2.

6 + \frac{10}{12} ≤ s(\overline{M}_{12}) := inf_{D∈Eff(\overline{M}_{12})}s(D) ≤ \frac{4415}{642} (= 6 + \frac{10}{12} + \frac{14}{321}).

Another immediate application, via [Log], [F1], concerns the birational type of \overline{M}_{g,n}:

Theorem 6.3. The moduli space of n-pointed curves \overline{M}_{12,n} is of general type for n ≥ 11.

The divisor 𝔇_{12} is constructed as the push-forward of a codimension 3 cycle in the stack 𝔊_{14}^{4} → M_{12} classifying linear series 𝔤_{14}^{4}. We describe the construction of this cycle, then extend this determinantal structure over a partial compactification of M_{12}. This will be essential to understand the intersection of \overline{𝔇}_{12} with the boundary divisors Δ_{0} and Δ_{1} of \overline{M}_{12}. We denote by M_{12}^{p} the open substack of M_{12} consisting of curves [C] ∈ M_{12} such that W_{13}^{4}(C) = ∅ and W_{14}^{5}(C) = ∅. Results in Brill-Noether theory guarantee that codim(M_{12} − M_{12}^{p}, M_{12}) ≥ 3. If 𝔓𝔦𝔠_{12}^{14} denotes the Picard stack of degree 14 over M_{12}^{p}, then we consider the smooth Deligne-Mumford substack 𝔊_{14}^{4} ⊂ 𝔓𝔦𝔠_{12}^{14} parameterizing pairs [C, L], where [C] ∈ M_{12}^{p} and L ∈ W_{14}^{4}(C) is a (necessarily complete and base point free) linear series. We denote by σ : 𝔊_{14}^{4} → M_{12}^{p} the forgetful morphism. For a general [C] ∈ M_{12}^{p}, the fibre σ^{−1}([C]) = W_{14}^{4}(C) is a smooth surface.

Let π : M_{12,1}^{p} → M_{12}^{p} be the universal curve and then p_{2} : M_{12,1}^{p} ×_{M}_{12}^{p} 𝔊_{14}^{4} → 𝔊_{14}^{4} denotes the natural projection. If L is a Poincaré bundle over M_{12,1}^{p} ×_{M}_{12}^{p} 𝔊_{14}^{4} (or over an étale cover), then by Grauert’s Theorem, both E := (p_{2})_{∗}(L) and F := (p_{2})_{∗}(L^{⊗2}) are vector bundles over 𝔊_{14}^{4}, with rank(E) = 5 and rank(F) = h^{0}(C, L^{⊗2}) = 17 respectively. There is a natural vector bundle morphism over 𝔊_{14}^{4} given by multiplication of sections,

ϕ : Sym^{2}(E) → F,

and we denote by U_{12} ⊂ 𝔊_{14}^{4} its first degeneracy locus. We set 𝔇_{12} := σ_{∗}(U_{12}). Since the degeneracy locus U_{12} has expected codimension 3 inside 𝔊_{14}^{4}, the locus 𝔇_{12} is a virtual divisor on M_{12}^{p}.

We extend the vector bundles E and F over a partial compactification of 𝔊_{14}^{4} given by limit 𝔤_{14}^{4}. We denote by Δ_{1}^{p} ⊂ Δ_{1} ⊂ \overline{M}_{12} the locus of curves [C ∪_{y} E], where E is an arbitrary elliptic curve, [C] ∈ M_{11} is a Brill-Noether general curve and y ∈ C is an arbitrary point. We then denote by Δ_{0}^{p} ⊂ Δ_{0} ⊂ \overline{M}_{12} the locus consisting of curves [C_{yq}] ∈ Δ_{0}, where [C, q] ∈ M_{11,1} is Brill-Noether general and y ∈ C is arbitrary, as well as their degenerations [C ∪_{q} E_{∞}] where E_{∞} is a rational nodal curve. Once we set

\overline{M}_{12}^{p} := M_{12}^{p} ∪ Δ_{0}^{p} ∪ Δ_{1}^{p} ⊂ \overline{M}_{12},

we can extend the morphism σ to a proper morphism

σ : 𝔊̃_{14}^{4} → \overline{M}_{12}^{p},

from the stack 𝔊̃_{14}^{4} of limit linear series 𝔤_{14}^{4} over the partial compactification \overline{M}_{12}^{p} of M_{12}.

We extend the vector bundles E and F over the stack 𝔊̃_{14}^{4}. The proof of the following result proceeds along the lines of the proof of Proposition 3.9 in [F1]:

Proposition 6.4. There exist two vector bundles E and F defined over 𝔊̃_{14}^{4} with rank(E) = 5 and rank(F) = 17, together with a vector bundle morphism ϕ : Sym^{2}(E) → F, such that the following statements hold:

• For [C, L] ∈ 𝔊_{14}^{4}, with [C] ∈ M_{12}^{p}, we have that
E(C, L) = H^{0}(C, L) and F(C, L) = H^{0}(C, L^{⊗2}).

• For t = (C ∪_{y} E, l_{C}, l_{E}) ∈ σ^{−1}(Δ_{1}^{p}), where g(C) = 11, g(E) = 1 and l_{C} = |L_{C}| is

such that L_{C} ∈ W_{14}^{4}(C) has a cusp at y ∈ C, then E(t) = H^{0}(C, L_{C}) and
F(t) = H^{0}(C, L_{C}^{⊗2}(−2y)) ⊕ ℂ · u^{2},

where u ∈ H^{0}(C, L_{C}) is any section such that ord_{y}(u) = 0. If L_{C} has a base point at y, then E(t) = H^{0}(C, L_{C}) = H^{0}(C, L_{C} ⊗ O_{C}(−y)) and the image of a natural map

F(t) → H^{0}(C, L_{C}^{⊗2}) is the subspace H^{0}(C, L_{C}^{⊗2} ⊗ O_{C}(−2y)).

• Fix t = [C_{yq} := C/y ∼ q, L] ∈ σ^{−1}(Δ_{0}^{p}), with q, y ∈ C and L ∈ \overline{W} _{14}^{4}(C_{yq}) such that h^{0}(C, ν^{∗}L ⊗ O_{C}(−y − q)) = 4, where ν : C → C_{yq} is the normalization map. In the case when L is locally free we have that

E(t) = H^{0}(C, ν^{∗}L) and F(t) = H^{0}(C, ν^{∗}L^{⊗2} ⊗ O_{C}(−y − q)) ⊕ ℂ · u^{2},

where u ∈ H^{0}(C, ν^{∗}L) is any section not vanishing at y and q. In the case when L is not locally free, that is, L ∈ \overline{W} _{14}^{4}(C_{yq}) − W_{14}^{4}(C_{yq}), then L = ν_{∗}(A), where A ∈ W_{13}^{4}(C) and the image of the natural map F(t) → H^{0}(C, ν^{∗}L^{⊗2}) is the subspace H^{0}(C, A^{⊗2}).

To determine the push-forward [\overline{𝔇}_{12}]^{virt} = σ_{∗}(c_{3}(F − Sym^{2}(E)) ∈ A^{1}(M_{12}^{p}), we study the restriction of the morphism ϕ along the pull-backs of two curves sitting in the boundary of \overline{M}_{12} and which are defined as follows: We fix a general pointed curve [C, q] ∈ M_{11,1} and a general elliptic curve [E, y] ∈ M_{1,1}. Then we consider the families

C_{0} := \{C/y ∼ q : y ∈ C\} ⊂ Δ_{0}^{p} ⊂ \overline{M}_{12} and C_{1} := \{C ∪_{y} E : y ∈ C\} ⊂ Δ_{1}^{p} ⊂ \overline{M}_{12}.

These curves intersect the generators of Pic(\overline{M}_{12}) as follows:

C_{0} · λ = 0, C_{0} · δ_{0} = deg(ω_{C_{yq}}) = −22, C_{0} · δ_{1} = 1 and C_{0} · δ_{j} = 0 for 2 ≤ j ≤ 6, and

C_{1} · λ = 0, C_{1} · δ_{0} = 0, C_{1} · δ_{1} = −deg(K_{C}) = −20 and C_{1} · δ_{j} = 0 for 2 ≤ j ≤ 6.

Next, we fix a general pointed curve [C, q] ∈ M_{11,1} and describe the geometry of the pull-back σ^{∗}(C_{0}) ⊂ 𝔊̃_{14}^{4}. We consider the determinantal 3-fold

Y := {(y, L) ∈ C × W_{14}^{4}(C) : h^{0}(C, L ⊗ O_{C}(−y − q)) = 4}

together with the projection π_{1} : Y → C. Inside Y we consider the following divisors

Γ_{1} := {(y, A ⊗ O_{C}(y)) : y ∈ C, A ∈ W_{13}^{4}(C)} and
Γ_{2} := {(y, A ⊗ O_{C}(q)) : y ∈ C, A ∈ W_{13}^{4}(C)}

intersecting transversally along the curve Γ := {(q, A ⊗ O_{C}(q)) : A ∈ W_{13}^{4}(C)} ≅ W_{13}^{4}(C). We introduce the blow-up Y′ → Y of Y along Γ and denote by E_{Γ} ⊂ Y′ the exceptional divisor and by Γ̃_{1}, Γ̃_{2} ⊂ Y′ the strict transforms of Γ_{1} and Γ_{2} respectively. We then define Ỹ := Y′/Γ̃_{1} ≅ Γ̃_{2}, to be the variety obtained from Y′ by identifying the divisors Γ̃_{1} and Γ̃_{2} over each (y, A) ∈ C × W_{13}^{4}(C). Let ϵ : Ỹ → Y be the projection map.

Proposition 6.5. With notation as above, one has a birational morphism of 3-folds

f : σ^{∗}(C_{0}) → Ỹ,

which is an isomorphism outside a curve contained in ϵ^{−1}(π_{1}^{−1}(q)). The map f_{|(π_{1}ϵf)^{−1}(q)} corresponds to forgetting the E_{∞}-aspect of each limit linear series. Accordingly, the vector bundles E_{|σ^{∗}(C_{0})} and F_{|σ^{∗}(C_{0})} are pull-backs under ϵ ∘ f of vector bundles on Y.

Proof. We fix a point y ∈ C − {q} and denote by ν : C → C_{yq} the normalization map, with ν(y) = ν(q). We investigate the variety \overline{W} _{14}^{4}(C_{yq}) ⊂ \overline{Pic}^{14}(C_{yq}) of torsion-free sheaves L on C_{yq} with deg(L) = 14 and h^{0}(C_{yq}, L) ≥ 5. A locally free L ∈ \overline{W} _{14}^{4}(C_{yq}) is determined by ν^{∗}(L) ∈ W_{14}^{4}(C), which has the property h^{0}(C, ν^{∗}L ⊗ O_{C}(−y − q)) = 4 (use that since W_{12}^{4}(C) = ∅, there exists a section of L that does not vanish simultaneously at both y and q). However, the line bundles of type A ⊗ O_{C}(y) or A ⊗ O_{C}(q) with A ∈ W_{13}^{4}(C), do not appear in this association, though (y, A ⊗ O_{C}(y)), (y, A ⊗ O_{C}(q)) ∈ Y. In fact, they correspond to the situation when L ∈ \overline{W} _{14}^{4}(C_{yq}) is not locally free, in which case necessarily L = ν_{∗}(A) for some A ∈ W_{13}^{4}(C). Thus, for a point y ∈ C − {q}, there is a birational morphism π_{1}^{−1}(y) → \overline{W} _{14}^{4}(C_{yq}) which is an isomorphism over the locus of locally free sheaves. More precisely, \overline{W} _{14}^{4}(C_{yq}) is obtained from π_{1}^{−1}(y) by identifying the disjoint divisors Γ_{1} ∩ π_{1}^{−1}(y) and Γ_{2} ∩ π_{1}^{−1}(y).

A special analysis is required when y = q, when C_{yq} degenerates to C ∪_{q} E_{∞}, where E_{∞} is a rational nodal cubic. If {l_{C}, l_{E_{∞}}} ∈ σ^{−1}([C ∪_{q} E_{∞}]), then the corresponding Brill-Noether numbers with respect to q satisfy ρ(l_{C}, q) ≥ 0 and ρ(l_{E_{∞}}, q) ≤ 2. The statement about the restrictions E_{|σ^{∗}(C_{0})} and F_{|σ^{∗}(C_{0})} follows, because both restrictions are defined by dropping the information coming from the elliptic tail. □

To describe σ^{∗}(C_{1}) ⊂ 𝔊̃_{14}^{4}, where [C] ∈ M_{11}, we define the determinantal 3-fold

X := {(y, L) ∈ C × W_{14}^{4}(C) : h^{0}(L ⊗ O_{C}(−2y)) = 4}.

In what follows we use notation from [EH1], to denote vanishing sequences of limit linear series:

Proposition 6.6. With notation as above, the 3-fold X is an irreducible component of σ^{∗}(C_{1}). Moreover one has that c_{3}((F − Sym^{2}E)_{|σ^{∗}(C_{1})}) = c_{3}((F − Sym^{2}E)_{|X}).

Proof. By the additivity of the Brill-Noether number, if {l_{C}, l_{E}} ∈ σ^{−1}([C∪_{y}E]), we have that 2 = ρ(12, 4, 14) ≥ ρ(l_{C}, y)+ρ(l_{E}, y). Since ρ(l_{E}, y) ≥ 0, we obtain that ρ(l_{C}, y) ≤ 2. If ρ(l_{E}, y) = 0, then l_{E} = 9y+|O_{E}(5y)|, that is, l_{E} is uniquely determined, while the aspect l_{C} ∈ G_{14}^{4}(C) is a complete 𝔤_{14}^{4} with a cusp at the variable point y ∈ C. This gives rise to an element from X. The remaining components of σ^{∗}(C_{1}) are indexed by Schubert indices ᾱ := (0 ≤ α_{0} ≤ ⋯ ≤ α_{4} ≤ 10) such that ᾱ > (0, 1, 1, 1, 1) and 5 ≤ ∑_{j=0}^{4} α_{j} ≤ 7. For such ᾱ, we set ᾱ^{c} := (10 − α_{4}, …, 10 − α_{0}) to be the complementary Schubert index, then define

Xᾱ := {(y, l_{C}) ∈ C × G_{14}^{4}(C) : α^{l_{C}}(y) ≥ ᾱ} and Zᾱ := {l_{E} ∈ G_{14}^{4}(E) : α^{l_{E}}(y) ≥ ᾱ^{c}}.

Then σ^{∗}(C_{1}) = X + ∑ᾱ Xᾱ × Zᾱ. The last claim follows by dimension reasons. Since dim Xᾱ = 1+ρ(11, 4, 14)−∑_{j=0}^{4} α_{j} < 3, for every ᾱ > (0, 1, 1, 1, 1) and the restrictions of both E and F are pulled-back from Xᾱ, one obtains that c_{3}(F − Sym^{2}E)_{|Xᾱ×Zᾱ} = 0. □

We also recall standard facts about intersection theory on Jacobians. For a Brill-Noether general curve [C] ∈ M_{g}, we denote by P a Poincaré bundle on C ×Pic^{d}(C) and by π_{1} : C × Pic^{d}(C) → C and π_{2} : C × Pic^{d}(C) → Pic^{d}(C) the projections. We define the cohomology class η = π_{1}^{∗}([point]) ∈ H^{2}(C × Pic^{d}(C)), and if δ_{1}, …, δ_{2g} ∈ H^{1}(C, ℤ) ≅ H^{1}(Pic^{d}(C), ℤ) is a symplectic basis, then we set

γ := − ∑_{α=1}^{g}(π_{1}^{∗}(δ_{α})π_{2}^{∗}(δ_{g+α}) − π_{1}^{∗}(δ_{g+α})π_{2}^{∗}(δ_{α})) ∈ H^{2}(C × Pic^{d}(C)).

One has the formula c_{1}(P) = dη + γ, corresponding to the Hodge decomposition of c_{1}(P), as well as the relations γ^{3} = 0, γη = 0, η^{2} = 0 and γ^{2} = −2ηπ_{2}^{∗}(θ). On W_{d}^{r}(C) there is a tautological rank r + 1 vector bundle M := (π_{2})_{∗}(P_{|C×W_{d}^{r}(C)}). To compute the Chern numbers of M we employ the Harris-Tu formula [HT]. We write ∑_{i=0}^{r} c_{i}(M^{∨}) = (1 + x_{1}) ⋯ (1 + x_{r+1}), and then for every class ζ ∈ H^{∗}(Pic^{d}(C), ℤ) one has the following formula:

(5)  x_{1}^{i_{1}} ⋯ x_{r+1}^{i_{r+1}} ζ = det(\frac{θ^{g+r−d+i_{j}−j+l}}{(g + r − d + i_{j} − j + l)!})_{1≤j,l≤r+1} ζ.

We compute the classes of the 3-folds that appear in Propositions 6.5 and 6.6:

Proposition 6.7. Let [C, q] ∈ M_{11,1} be a Brill-Noether general pointed curve. If M denotes the tautological rank 5 vector bundle over W_{14}^{4}(C) and c_{i} := c_{i}(M^{∨}) ∈ H^{2i}(W_{14}^{4}(C), ℂ), then one has the following relations:

(i) [X] = π_{2}^{∗}(c_{4}) − 6ηθπ_{2}^{∗}(c_{2}) + (48η + 2γ)π_{2}^{∗}(c_{3}) ∈ H^{8}(C × W_{14}^{4}(C), ℂ).

(ii) [Y] = π_{2}^{∗}(c_{4}) − 2ηθπ_{2}^{∗}(c_{2}) + (13η + γ)π_{2}^{∗}(c_{3}) ∈ H^{8}(C × W_{14}^{4}(C), ℂ).

Proof. We start by noting that W_{14}^{4}(C) is a smooth 6-fold isomorphic to the symmetric product C_{6}. We realize X as the degeneracy locus of a vector bundle morphism defined over C × W_{14}^{4}(C). For each pair (y, L) ∈ C × W_{14}^{4}(C), there is a natural map

H^{0}(C, L ⊗ O_{2y})^{∨} → H^{0}(C, L)^{∨}

which globalizes to a vector bundle morphism ζ : J_{1}(P)^{∨} → π_{2}^{∗}(M)^{∨} over C × W_{14}^{4}(C). Then we have the identification X = Z_{1}(ζ) and the Thom-Porteous formula gives that

[X] = c_{4}(π_{2}^{∗}(M) − J_{1}(P^{∨})). From the usual exact sequence over C × Pic^{14}(C)

0 ⟶ π_{1}^{∗}(K_{C}) ⊗ P ⟶ J_{1}(P) ⟶ P ⟶ 0,

we can compute the total Chern class of the jet bundle

c_{t}(J_{1}(P)^{∨})^{−1} = (∑_{j≥0}(d(L)η+γ)^{j})·(∑_{j≥0}((2g(C)−2+d(L))η+γ)^{j}) = 1−6ηθ+48η+2γ,

which quickly leads to the formula for [X]. To compute [Y] we proceed in a similar way. We denote by μ, ν : C × C × Pic^{14}(C) → C × Pic^{14}(C) the two projections, by Δ ⊂ C × C × Pic^{14}(C) the diagonal and we set Γ_{q} := {q} × Pic^{14}(C). We introduce the rank 2 vector bundle B := (μ)_{∗}(ν^{∗}(P) ⊗ O_{Δ+ν^{∗}(Γ_{q})}) defined over C × W_{14}^{4}(C). We note that there is a bundle morphism χ : B^{∨} → (π_{2})^{∗}(M)^{∨}, such that Y = Z_{1}(χ). Since we also have that

c_{t}(B^{∨})^{−1} = (1 + (d(L)η + γ) + (d(L)η + γ)^{2} + ⋯)(1 − η),

we immediately obtained the stated expression for [Y]. □

Proposition 6.8. Let [C] ∈ M_{11} and denote by μ, ν : C × C × Pic^{14}(C) → C × Pic^{14}(C) the natural projections. We define the vector bundles A_{2} and B_{2} on C × Pic^{14}(C) having fibres

A_{2}(y, L) = H^{0}(C, L^{⊗2} ⊗ O_{C}(−2y)) and B_{2}(y, L) = H^{0}(C, L^{⊗2} ⊗ O_{C}(−y − q)),

respectively. One has the following formulas:

c_{1}(A_{2}) = −4θ − 4γ − 76η c_{1}(B_{2}) = −4θ − 2γ − 27η,
c_{2}(A_{2}) = 8θ^{2} + 280ηθ + 16γθ, c_{2}(B_{2}) = 8θ^{2} + 100ηθ + 8θγ,

c_{3}(A_{2}) = −\frac{32}{3}θ^{3} − 512ηθ^{2} − 32θ^{2}γ and c_{3}(B_{2}) = −\frac{32}{3}θ^{3} − 184ηθ^{2} − 16θ^{2}γ.

Proof. Immediate application of Grothendieck-Riemann-Roch with respect to ν. □

Before our next result, we recall that if V is a vector bundle of rank r + 1 on a variety X, we have the formulas:

(i) c_{1}(Sym^{2}(V)) = (r + 2)c_{1}(V).

(ii) c_{2}(Sym^{2}(V)) = \frac{r(r+3)}{2}c_{1}^{2}(V) + (r + 3)c_{2}(V).
(iii) c_{3}(Sym^{2}(V)) = \frac{r(r+4)(r−1)}{6}c_{1}^{3}(V) + (r + 5)c_{3}(V) + (r^{2} + 4r − 1)c_{1}(V)c_{2}(V).
We expand σ_{∗}(c_{3}(F − Sym^{2}E)) ≡ aλ − b_{0}δ_{0} − b_{1}δ_{1} ∈ A^{1}(M_{12}^{p}) and determine the

coefficients a, b_{0} and b_{1}. This will suffice in order to compute s(\overline{𝔇}_{12}).

Theorem 6.9. Let [C] ∈ M_{11} be a Brill-Noether general curve and denote by C_{1} ⊂ Δ_{1} ⊂ \overline{M}_{12} the associated test curve. Then the coefficient of δ_{1} in the expansion of \overline{𝔇}_{22} is equal to

b_{1} = \frac{1}{2g(C) − 2}σ^{∗}(C_{1}) · c_{3}(F − Sym^{2}E) = 9867.

Proof. We intersect the degeneracy locus of the map ϕ : Sym^{2}(E) → F with the 3-fold σ^{∗}(C_{1}) = X + ∑ᾱ Xᾱ × Zᾱ. As already explained in Proposition 6.6, it is enough to estimate the contribution coming from X and we can write

σ^{∗}(C_{1}) · c_{3}(F − Sym^{2}E) = c_{3}(F_{|X}) − c_{3}(Sym^{2}E_{|X}) − c_{1}(F_{|X})c_{2}(Sym^{2}E_{|X})+

+2c_{1}(Sym^{2}E_{|X})c_{2}(Sym^{2}E_{|X})−c_{1}(Sym^{2}E_{|X})c_{2}(F_{|X})+c_{1}^{2}(Sym^{2}E_{|X})c_{1}(F_{|X})−c_{1}^{3}(Sym^{2}E_{|X}). We are going to compute each term in the right-hand-side of this expression.

Recall that we have constructed in Proposition 6.7 a vector bundle morphism ζ : J_{1}(P)^{∨} → π_{2}^{∗}(M)^{∨}. We consider the kernel line bundle Ker(ζ). If U is the line bundle on X with fibre

U(y, L) = \frac{H^{0}(C, L)}{H^{0}(C, L ⊗ O_{C}(−2y))} ↪ H^{0}(C, L ⊗ O_{2y})

over a point (y, L) ∈ X, then one has an exact sequence over X

0 → U → J_{1}(P) → Ker(ζ)^{∨} → 0.

In particular, c_{1}(U) = 2γ +48η−c_{1}(Ker(ζ))^{∨}. The products of the Chern class of Ker(ζ)^{∨} with other classes on C × W_{14}^{4}(C) can be computed from the Harris-Tu formula [HT]: (6) c_{1}(Ker(ζ)^{∨})·ξ_{|X} = −c_{5}(π_{2}^{∗}(M)^{∨}−J_{1}(P)^{∨})·ξ_{|X} = −(π_{2}^{∗}(c_{5})−6ηθπ_{2}^{∗}(c_{3})+(48η+2γ)π_{2}^{∗}(c_{4}))·ξ_{|X},

for any class ξ ∈ H^{2}(C × W_{14}^{4}(C), ℂ).

If A_{3} denotes the rank 18 vector bundle on X having fibres A_{3}(y, L) = H^{0}(C, L^{⊗2}), then there is an injective morphism U^{⊗2} ↪ A_{3}/A_{2}, and we consider the quotient sheaf

G := \frac{A_{3}/A_{2}}{U^{⊗2}} .

Since the morphism U^{⊗2} → A_{3}/A_{2} vanishes along the locus of pairs (y, L) where L has a base point, G has torsion along Γ ⊂ X. A straightforward local analysis now shows that F_{|X} can be identified as a subsheaf of A_{3} with the kernel of the map A_{3} → G. Therefore, there is an exact sequence of vector bundles on X

0 → A_{2|X} → F_{|X} → U^{⊗2} → 0,

which over a general point of X corresponds to the decomposition

F(y, L) = H^{0}(C, L^{⊗2} ⊗ O_{C}(−2y)) ⊕ ℂ · u^{2},

where u ∈ H^{0}(C, L) is such that ord_{y}(u) = 1. The analysis above, shows that the sequence stays exact over the curve Γ as well. Hence

c_{1}(F_{|X}) = c_{1}(A_{2|X}) + 2c_{1}(U), c_{2}(F_{|X}) = c_{2}(A_{2|X}) + 2c_{1}(A_{2|X})c_{1}(U) and
c_{3}(F_{|X}) = c_{3}(A_{2}) + 2c_{2}(A_{2|X})c_{1}(U).

Furthermore, since E_{|X} = π_{2}^{∗}(M)_{|X}, we obtain that:

σ^{∗}(C_{1}) · c_{3}(F − Sym^{2}E) = c_{3}(A_{2|X}) + c_{2}(A_{2|X})c_{1}(U^{⊗2}) − c_{3}(Sym^{2}π_{2}^{∗}M_{|X})−

−(\frac{r(r + 3)}{2}c_{1}(π_{2}^{∗}M_{|X})+(r+3)c_{2}(π_{2}^{∗}M_{|X}))·(c_{1}(A_{2|X})+c_{1}(U^{⊗2})−2(r+2)c_{1}(π_{2}^{∗}M_{|X}))−

−(r + 2)c_{1}(π_{2}^{∗}M_{|X})c_{2}(A_{2|X}) − (r + 2)c_{1}(π_{2}^{∗}M_{|X})c_{1}(A_{2|X})c_{1}(U^{⊗2})+

+(r + 2)^{2}c_{1}^{2}(π_{2}^{∗}M_{|X})c_{1}(A_{2|X}) + (r + 2)^{2}c_{1}^{2}(π_{2}^{∗}M_{|X})c_{1}(U^{⊗2}) − (r + 2)^{3}c_{1}^{3}(π_{2}^{∗}M_{|X}).

As before, c_{i}(π_{2}^{∗}M_{|X}^{∨}) = π_{2}^{∗}(c_{i}) ∈ H^{2i}(X, ℂ). The coefficient of c_{1}(Ker(ζ)^{∨}) in the product σ^{∗}(C_{1}) · c_{3}(F − Sym^{2}E) is evaluated via (6). The part of this product that does not contain c_{1}(Ker(ζ)^{∨}) equals

28π_{2}^{∗}(c_{2})θ − 88π_{2}^{∗}(c_{1}^{2})θ + 440ηπ_{2}^{∗}(c_{1}^{2}) − 53π_{2}^{∗}(c_{1}c_{2}) − \frac{32}{3}θ^{3} + 128ηθ^{2} − 432ηθπ_{2}^{∗}(c_{1})

+64π_{2}^{∗}(c_{1}^{3}) − 140ηπ_{2}^{∗}(c_{2}) + 48θ^{2}π_{2}^{∗}(c_{1}) + 9π_{2}^{∗}(c_{3}) ∈ H^{6}(C × W_{14}^{4}(C), ℂ).

Multiplying this quantity by the class [X] obtained in Proposition 6.7 and then adding to it the contribution coming from c_{1}(Ker(ζ)^{∨}), one obtains a homogeneous polynomial of degree 7 in η, θ and π_{2}^{∗}(c_{i}) for 1 ≤ i ≤ 4. The only non-zero monomials are those containing η. After retaining only these monomials, the resulting degree 6 polynomial in θ, c_{i} ∈ H^{∗}(W_{14}^{4}(C), ℤ) can be brought to a manageable form, by noting that, since h^{1}(C, L) = 1, the classes c_{i} are not independent. Precisely, if one fixes a divisor D ∈ C_{e} of large degree, there is an exact sequence

0 → M → (π_{2})_{∗}(P ⊗ O(π^{∗}D)) → (π_{2})_{∗}(P ⊗ O(π_{1}^{∗}D)_{|π_{1}^{∗}D}) → R^{1}π_{2∗}(P_{|C×W_{14}^{4}(C)}) → 0,

from which, via the well-known fact c_{t}((π_{2})_{∗}(P ⊗ O(π_{1}^{∗}D))) = e^{−θ}, it follows that

c_{t}R^{1}π_{2∗}(P_{|C×W_{14}^{4}(C)}) · e^{−θ} = ∑_{i=0}^{4}(−1)^{i}c_{i}.

Hence c_{i+1} = θ^{i}c_{i}/i!−iθ^{i+1}/(i+1)!, for all i ≥ 2. After routine manipulations, one finds that b_{1} = σ^{∗}(C_{1}) · c_{3}(F − Sym^{2}(E))/20 = 9867. □

Theorem 6.10. Let [C, q] ∈ M_{11,1} be a Brill-Noether general pointed curve and we denote by C_{0} ⊂ Δ_{0} ⊂ \overline{M}_{12} the associated test curve. Then σ^{∗}(C_{0})·c_{3}(F −Sym^{2}E) = 22b_{0}−b_{1} = 32505. It follows that b_{0} = 1926.

Proof. As already noted in Proposition 6.5, the vector bundles E_{|σ^{∗}(C_{0})} and F_{|σ^{∗}(C_{0})} are both pull-backs of vector bundles on Y and we denote these vector bundles E and F as well, that is, E_{|σ^{∗}(C^{0})} = (ϵ ∘ f)^{∗}(E_{|Y}) and F_{|σ^{∗}(C_{0})} = (ϵ ∘ f)^{∗}(F_{|Y}). Like in the proof of Theorem 6.9, we evaluate each term appearing in σ^{∗}(C_{0}) · c_{3}(F − Sym^{2}(E)).

Let V be the line bundle on Y with fibre

V (y, L) = \frac{H^{0}(C, L)}{H^{0}(C, L ⊗ O_{C}(−y − q))} ↪ H^{0}(C, L ⊗ O_{y+q})

over a point (y, L) ∈ Y. There is an exact sequence of vector bundles over Y

0 ⟶ V ⟶ B ⟶ Ker(χ)^{∨} ⟶ 0,

where χ : B^{∨} → π_{2}^{∗}(M)^{∨} is the bundle morphism defined in the second part of Proposition 6.7. In particular, c_{1}(V) = 13η + γ − c_{1}(Ker(χ^{∨}). By using again [HT], we find the following formulas for the Chern numbers of Ker(χ)^{∨}:

c_{1}(Ker(χ)^{∨})·ξ_{|Y} = −c_{5}(π_{2}^{∗}(M)^{∨}−B^{∨})·ξ_{|Y} = −(π_{2}^{∗}(c_{5})+π_{2}^{∗}(c_{4})(13η+γ)−2π_{2}^{∗}(c_{3})ηθ)·ξ_{|Y} ,

for any class ξ ∈ H^{2}(C × W_{14}^{4}(C), ℂ). Recall that we introduced the vector bundle B_{2} over C × W_{14}^{4}(C) with fibre B_{2}(y, L) = H^{0}(C, L^{⊗2} ⊗ O_{C}(−y − q)). We claim that one has an exact sequence of bundles over Y

(7)  0 ⟶ B_{2|Y} ⟶ F_{|Y} ⟶ V^{⊗2} ⟶ 0.

If B_{3} is the vector bundle on Y with fibres B_{3}(y, L) = H^{0}(C, L^{⊗2}), we have an injective morphism of sheaves V^{⊗2} ↪ B_{3}/B_{2} locally given by

v^{⊗2} ↦ v^{2} mod H^{0}(C, L^{⊗2} ⊗ O_{C}(−y − q)),

where v ∈ H^{0}(C, L) is any section not vanishing at q and y. Then F_{|Y} is canonically identified with the kernel of the projection morphism

B_{3} → \frac{B_{3}/B_{2}}{V^{⊗2}}

and the exact sequence (7) now becomes clear. Therefore c_{1}(F_{|Y}) = c_{1}(B_{2|Y}) + 2c_{1}(V), c_{2}(F_{|Y}) = c_{2}(B_{2|Y}) + 2c_{1}(B_{2|Y})c_{1}(V) and c_{3}(F_{|Y}) = c_{3}(B_{2|Y}) + 2c_{2}(B_{2|Y})c_{1}(V). The part of the total intersection number σ^{∗}(C_{0}) · c_{3}(F − Sym^{2}(E)) that does not contain c_{1}(Ker(χ^{∨})) equals

28π_{2}^{∗}(c_{2})θ − 88π_{2}^{∗}(c_{1}^{2})θ − 22ηπ_{2}^{∗}(c_{1}^{2}) − 53π_{2}^{∗}(c_{1}c_{2}) − \frac{32}{3}θ^{3}+

−8ηθ^{2} + 24ηθπ_{2}^{∗}(c_{1}) + 64π_{2}^{∗}(c_{1}^{3}) + 7ηπ_{2}^{∗}(c_{2}) + 48θ^{2}π_{2}^{∗}(c_{1}) + 9π_{2}^{∗}(c_{3}) ∈ H^{6}(C × W_{14}^{4}(C), ℂ) and this gets multiplied with the class [Y] from Proposition 6.7. The coefficient of c_{1}(Ker(ζ)^{∨}) in σ^{∗}(C_{0}) · c_{3}(F − Sym^{2}E) equals

−2c_{2}(B_{2|Y}) − 2(r + 2)^{2}π_{2}^{∗}(c_{1}^{2}) − 2(r + 2)c_{1}(B_{2|Y})π_{2}^{∗}(c_{1}) + r(r + 3)π_{2}^{∗}(c_{1}^{2}) + 2(r + 3)π_{2}^{∗}(c_{2}).

All in all, 22b_{0} − b_{1} = σ^{∗}(C_{0}) · c_{3}(F − Sym^{2}E) and we evaluate this using (6). □

The following result follows from the definition of the vector bundles E and F given in Proposition 6.4:

Theorem 6.11. Let [C, q] ∈ M_{11,1} be a Brill-Noether general pointed curve and R ⊂ \overline{M}_{12} the pencil obtained by attaching at the fixed point q ∈ C a pencil of plane cubics. Then

a − 12b_{0} + b_{1} = σ_{∗}c_{3}(F − Sym^{2}E) · R = 0.

End of the proof of Theorem 6.1. The fact that the virtual divisor 𝔇_{12} is a genuine divisor on M_{12} follows from [T]. Assuming by contradiction that for every curve [C] ∈ M_{12}, there exists L ∈ W_{14}^{4}(C) such that μ_{0}(L) is not-injective, one can construct a stable vector bundle E of rank 2 sitting in an extension

0 ⟶ K_{C} ⊗ L^{∨} ⟶ E ⟶ L ⟶ 0,

such that h^{0}(C, E) = h^{0}(C, L) + h^{1}(C, L) = 7, and for which the Mukai-Petri map Sym^{2}H^{0}(C, E) → H^{0}(C, Sym^{2}E) is not injective. This is a contradiction. To determine the slope of the divisor \overline{D}_{12}, we write \overline{𝔇}_{12} ≡ aλ − ∑_{j=0}^{6} b_{j}δ_{j} ∈ Pic(\overline{M}_{12}). Since a/b_{0} = 4415/642 ≤ 71/10, we are in a position to apply Corollary 1.2 from [FP], which gives the inequalities b_{j} ≥ b_{0} for 1 ≤ j ≤ 6. Therefore s(\overline{𝔇}_{12}) = a/b_{0} < 13/2. □

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Humboldt-Universität zu Berlin, Institut Für Mathematik, Unter den Linden 6 10099 Berlin, Germany

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

Universitá Roma Tre, Dipartimenti di Matematica, Largo San Leonardo Murialdo 1-00146 Roma, Italy

E-mail address: verra@mat.unirom3.it