\overline{M}_{22} IS OF GENERAL TYPE

GAVRIL FARKAS

1. Introduction

The aim of this paper is to prove the following result:

Theorem: The moduli space of curves of genus 22 is of general type.

We recall that it is a famous result due to Harris, Mumford and Eisenbud that \overline{M}_{g} is of general type for g ≥ 24 (cf. [HM], [H1], [EH]). On the other hand, a classical theorem due to Severi says \overline{M}_{g} is unirational for g ≤ 10 (see [AC1] for a modern exposition). Similar results for higher g were proved in the 1980’s: Sernesi showed that \overline{M}_{12} is unirational (cf. [Se]), then Chang and Ran settled the unirationality of \overline{M}_{11} and \overline{M}_{13} and gave a different proof of Sernesi’s result in genus 12 (cf. [CR1]). One also knows that \overline{M}_{15} and \overline{M}_{16} have negative Kodaira dimension (cf. [CR2]). The case of \overline{M}_{14} remained open for a long time until recently when Verra proved that \overline{M}_{14} is also unirational (cf. [V]). The highest genus not entirely covered by [EH] is g = 23. One knows that the Kodaira dimension of \overline{M}_{23} is ≥ 2 (cf. [F1]).

Closely related to the problem of determining the Kodaira dimension of \overline{M}_{g} is the Harris-Morrison Slope Conjecture which asserts that the slope of every effective divisor on \overline{M}_{g} is ≥ 6 + 12/(g + 1) -this quantity being the slope of every Brill-Noether divisor \overline{M}_{g,d}^{r} of curves [C] ∈ M_{g} carrying a 𝔤_{d}^{r} when g − (r + 1)(g − d + r) = −1 (cf. [HMo], [EH]). Recalling that the canonical divisor on \overline{M}_{g} is given by the formula K_{\overline{M}_{g}} ≡ 13λ − 2δ_{0} − 3δ_{1} − 2δ_{2} − ⋯ − 2δ_{[g/2]}, the Slope Conjecture trivially implies that κ(\overline{M}_{g}) = −∞ for all g ≤ 22. However, the Slope Conjecture fails for a large number of genera (see [FP], [F2], [F3], [Kh]) in the sense that there are examples of effective divisors on \overline{M}_{g} of slope < 6 + 12/(g + 1). This raised the prospect of constructing an effective divisor D on \overline{M}_{g} for some g ≤ 23, having slope s(D) < 13/2 = s(K_{\overline{M}_{g}}). In [F2] we came very close to succeeding in this when we showed that on \overline{M}_{22}, the slope of the closure of the divisor Z_{22,2} consisting of curves [C] ∈ M_{22} for which there exists

a linear series L ∈ W_{30}^{10}(C) such that C ↪^{|L|} P^{10} fails to satisfy the Green-Lazarsfeld property (N_{2}), is equal to s(\overline{Z}_{22,2}) = 1665/256 = 6.503 … < 6 + 12/23.

In this paper we construct an effective divisor on \overline{M}_{22} of slope < 13/2 and prove the following result:

Research partially supported by an Alfred P. Sloan Fellowship, the NSF Grants DMS-0450670 and DMS-0500747 and a 2006 Texas Summer Research Assignment.

Theorem 1.1. The following locus of smooth curves of genus 22

𝔇_{22} := {[C] ∈ M_{22} : ∃L ∈ W_{25}^{6}(C) withSym^{2}H^{0}(C, L) → H^{0}(C, L^{⊗2}) not injective}

is a divisor on M_{22} and the class of its compactification on \overline{M}_{22} is given by the formula:

\overline{𝔇}_{22} ≡ 132822768(\frac{17121}{2636}λ − δ_{0} − \frac{14511}{2636}δ_{0} − ∑_{j=2}^{11} b_{j}δ_{j}),

where b_{j} > 1 for 2 ≤ j ≤ 11. It follows that s(\overline{𝔇}_{22}) = 17121/2636 = 6.49506 …, hence \overline{M}_{22} is of general type.

2. The divisor 𝔇_{22}

In this section we construct two tautological vector bundles over the Severi variety of plane curves of genus 22 and degree 17 and define the divisor 𝔇_{22} as the image of the first degeneration locus of a natural map between these bundles.

We denote by M_{22}^{0} the open substack of M_{22} consisting of curves [C] ∈ M_{22} such that W_{24}^{6}(C) = ∅ and W_{25}^{7}(C) = ∅. Standard results in Brill-Noether theory guarantee that codim(M_{22} − M_{22}^{0}, M_{22}) ≥ 3. If 𝔓𝔦𝔠_{22}^{25} denotes the Picard stack of degree 25 over M_{22}^{0} (that is, theétale sheafification of the Picard functor), then we consider the substack 𝔊_{25}^{6} ⊂ 𝔓𝔦𝔠_{22}^{25} parametrizing pairs [C, L] where [C] ∈ M_{22}^{0} and L ∈ W_{25}^{6}(C). We denote by σ : 𝔊_{25}^{6} → M_{22}^{0} the forgetful morphism. Note that if L ∈ W_{25}^{6}(C) then K_{C} ⊗ L^{∨} ∈ W_{17}^{2}(C) and since the classical Severi variety of plane curves of given degree and genus is irreducible (cf. [H2]), it follows that 𝔊_{25}^{6} is irreducible as well. For a general [C] ∈ M_{22}^{0}, the fibre σ^{−1}([C]) = W_{25}^{6}(C) is a smooth curve and 𝔊_{25}^{6} is an irreducible stack of dimension dim(𝔊_{25}^{6}) = dim(M_{22}) + 1. Moreover, 𝔊_{25}^{6} is smooth at a point [C, L] if an

only if H^{1}(C, N_{f}) = 0, where N_{f} is normal line bundle of the map f : ⟶^{|K_{C}⊗L^{∨}|} P^{2}. As explained in [AC2], Proposition 2.9, the vanishing condition H^{1}(C, N_{f}) = 0 is satisfied outside a subset of 𝔊_{25}^{6} of dimension ≤ g−8 = 14, hence for the purpose of codimension ≤ 2 calculations we carry out in this paper, we can work with 𝔊_{25}^{6} as if it was a smooth stack.

We denote by π : M_{22,1}^{0} → M_{22}^{0} the universal curve over the moduli stack and by p_{2} : M_{22,1}^{0} ×_{M_{22}^{0}} 𝔊_{25}^{6} → 𝔊_{25}^{6} the natural projection. If L is the Poincaré bundle over M_{22,1}^{0} ×_{M_{22}^{0}} 𝔊_{25}^{6}, then by Grauert’s Theorem E := (p_{2})_{∗}(L) and F := (p_{2})_{∗}(L^{⊗2}) are vector bundles over 𝔊_{25}^{6} with rank(E) = 7 and rank(F) = 29. There is a natural vector bundle map ϕ : Sym^{2}(E) → F and we denote by U_{22} ⊂ 𝔊_{25}^{6} its first degeneracy locus. We set 𝔇_{22} := σ_{∗}(U_{22}) and clearly U_{22} has expected codimension 2 inside 𝔊_{25}^{6} hence 𝔇_{22} is a virtual divisor on M_{22}.

We shall extend the vector bundles E and F over a partial compactification of 𝔊_{25}^{6}. We denote by Δ_{1}^{0} ⊂ Δ_{1} ⊂ \overline{M}_{22} the locus of curves [C ∪_{y} E], where E is an arbitrary elliptic curve, [C] ∈ M_{21} is a Brill-Noether general curve of genus 21 and y ∈ C is an arbitrary point. We also denote by Δ_{0}^{0} ⊂ Δ_{0} ⊂ \overline{M}_{22} the locus consisting of curves C/y ∼ q, where [C, q] ∈ M_{21,1} is Brill-Noether general and y ∈ C is arbitary, as well as their degenerations [C ∪_{q} E_{∞}] where E_{∞} is a rational nodal curve (that is, j(E_{∞}) = ∞). Once we set M̃_{22} := M_{22}^{0} ∪Δ_{0}^{0} ∪Δ_{1}^{0}, one can extend the map σ : 𝔊_{25}^{6} → M_{22}^{0} to a proper map σ : 𝔊̃_{25}^{6} → M̃_{22} from the variety 𝔊̃_{25}^{6} of (generalized) limit linear series 𝔤_{25}^{6} over the tree-like curves from M̃_{22}.

Like in [F2], [F3], our technique for determining the class of the divisor \overline{𝔇}_{22} is to intersect it with two standard test curves sitting in the boundary of \overline{M}_{22}: we fix a general pointed curve [C, q] ∈ M_{21,1} and a general elliptic curve [E, y] ∈ M_{1,1}. Then we define the families

C^{0} := \{C/y ∼ q : y ∈ C\} ⊂ Δ_{0} ⊂ \overline{M}_{22} and C^{1} := \{C ∪_{y} E : y ∈ C\} ⊂ Δ_{1} ⊂ \overline{M}_{22}.

These curves intersect the generators of Pic(\overline{M}_{22}) as follows:

C^{0} · λ = 0, C^{0} · δ_{0} = −42, C^{0} · δ_{1} = 1 and C^{0} · δ_{j} = 0 for 2 ≤ j ≤ 11, and

C^{1} · λ = 0, C^{1} · δ_{0} = 0, C^{1} · δ_{1} = −40 and C^{1} · δ_{j} = 0 for 2 ≤ j ≤ 11.

Before we state the next results, we recall that if X is a stable curve whose dual graph is a tree and l is a limit 𝔤_{d}^{r} on X, for an irreducible component Y of X, we denote by l_{Y} = (L_{Y} , V_{Y} ⊂ H^{0}(L_{Y})) the Y -aspect of l. For y ∈ Y we denote by {a_{s}^{l_{Y}} (C)}_{s=0...r} the vanishing sequence of l at y and by ρ(l_{Y} , y) := ρ(g, r, d)−∑_{i=0}^{r}(a_{i}^{l_{Y}} (y)−i) the adjusted Brill-Noether number with respect to the point y.

Proposition 2.1. Fix general curves [C] ∈ M_{21} and [E, y] ∈ M_{1,1} and consider the associated test curve C^{1} ⊂ Δ_{1} ⊂ \overline{M}_{22}. Then we have the following equality of 2-cycles in 𝔊̃_{25}^{6}:

σ^{∗}(C^{1}) = X + X_{1} × X_{2} + Γ_{0} × Z_{0} + n_{1} · Z_{1} + n_{2} · Z_{2} + n_{3} · Z_{3},

where

X := {(y, L) ∈ C × W_{25}^{6}(C) : h^{0}(L ⊗ O_{C}(−2y)) = 6},
X_{1} := {(y, L) ∈ C × W_{25}^{6}(C) : a^{L}(y) = (0, 2, 3, 4, 5, 6, 8)},

X_{2} := \{l_{E} ∈ G_{8}^{6}(E) : a_{1}^{l_{E}}(y) ≥ 2, a_{6}^{l_{E}}(y) = 8\} ≅ P(\frac{H^{0}(O_{E}(8y))}{H^{0}(O_{E}(6y))})

Γ_{0} := {(y, A ⊗ O_{C}(y)) : y ∈ C, A ∈ W_{24}^{6}(C)}, Z_{0} = G_{7}^{6}(E) ≅ E,

Z_{1} := \{l_{E} ∈ G_{9}^{6}(E) : a_{1}^{l_{E}}(y) ≥ 3, a_{6}^{l_{E}}(y) = 9\} ≅ P(\frac{H^{0}(O_{E}(9y))}{H^{0}(O_{E}(6y))}),

Z_{2} := \{l_{E} ∈ G_{8}^{6}(E) : a_{2}^{l_{E}}(y) ≥ 3, a_{6}^{l_{E}}(y) = 8\} ≅ P(\frac{H^{0}(O_{E}(8y))}{H^{0}(O_{E}(5y))}),

Z_{3} := \{l_{E} ∈ G_{8}^{6}(E) : a^{l_{E}}(y) ≥ (0, 2, 3, 4, 5, 6, 7)\} ≅ ⋃_{z∈E} P(\frac{H^{0}(O_{E}(7y + z))}{H^{0}(O_{E}(5y + z))}),

where the constants n_{1}, n_{2} and n_{3} are explicitly known positive integers.

Remark 2.2. The constants n_{i}, 1 ≤ i ≤ 3 have the following enumerative interpretation. First n_{1} is the number of linear series L ∈ W_{25}^{6}(C) such that there exists an unspecified point y ∈ C with a^{L}(y) = (0, 2, 3, 4, 5, 6, 9). Similarly, n_{2} is the number of those L ∈ W_{25}^{6}(C) for which there exists y ∈ C with a^{L}(y) = (0, 2, 3, 4, 5, 7, 8). Finally n_{3} is the number of points y ∈ C such that there exists L ∈ W_{24}^{6}(C) which is ramified at y. If n_{0} is the number of 𝔤_{24}^{6}’s on C, then Γ_{0} consists of n_{0} disjoint copies of the curve C.

Proof. By the additivity of the Brill-Noether number, if {l_{C}, l_{E}} is a limit 𝔤_{25}^{6} on C ∪_{y} E, we have that 1 = ρ(22, 6, 25) ≥ ρ(l_{C}, y) + ρ(l_{E}, y). Since ρ(l_{E}, y) ≥ 0, we obtain that ρ(l_{C}, y) ≤ 1. If ρ(l_{E}, y) = 0, then l_{E} = 18y + |O_{E}(7y)|, that is, l_{E} is uniquely determined, while the C-aspect l_{C} is a complete 𝔤_{25}^{6} with a cusp at the variable point y ∈ C. This gives rise to an element from X. In the case when ρ(l_{C}, y) = 0 and ρ(l_{E}, y) = 1, we obtain that the underlying line bundle L_{C} of l_{C} either has a base point at y and then (y, L_{C}) ∈ Γ_{0}, or else, L_{C} belongs to the curve X_{1} and l_{E}(−17y) is a 𝔤_{8}^{6} on E having vanishing sequence ≥ (0, 2, 3, 4, 5, 6, 8), that is, l_{E}(−17y) is an element of X_{2}. Finally, we have to consider the case when ρ(l_{C}, y) = −1 and ρ(l_{E}, y) = 2. There are a finite number of such points y ∈ C, and running through all the possibilities we obtain the components Z_{i} for 1 ≤ i ≤ 3. □

Before stating our next result we introduce some notation. We fix a general pointed curve [C, q] ∈ M_{21,1} and denote by Y the following surface:

Y := {(y, L) ∈ C × W_{25}^{6}(C) : h^{0}(C, L ⊗ O_{C}(−y − q)) = 6}

and by π_{1} : Y → C the first projection. Inside Y we consider two curves corresponding to 𝔤_{25}^{6}’s with a base point at q:

Γ_{1} := {(y, A⊗O_{C}(y)) : y ∈ C, A ∈ W_{24}^{6}(C)} and Γ_{2} := {(y, A⊗O_{C}(q)) : y ∈ C, A ∈ W_{24}^{6}(C)}

intersecting transversally in n_{0} = |W_{24}^{6}(C)| points (Note that sine [C] ∈ M_{21} is Brill-Noether general, W_{24}^{6}(C) is a reduced 0-dimensional scheme consisting of n_{0} very ample (in particular, base point free) 𝔤_{24}^{6}’s). We denote by Y′ the blow-up of Y at these n_{0} points and at the points (q, B) ∈ Y where B ∈ W_{25}^{6}(C) is a linear series with the property that h^{0}(C, B ⊗ O_{C}(−8q)) ≥ 1. We denote by E_{A}, E_{B} ⊂ Y′ the exceptional divisors corresponding to (q, A ⊗ O_{C}(q)) and (q, B) respectively, by ϵ : Y′ → Y the projection and by Γ̃_{1}, Γ̃_{2} ⊂ Y′ the strict transforms of Γ_{1} and Γ_{2}.

Proposition 2.3. Fix a general curve [C, q] ∈ M_{21,1} and consider the associated test curve C^{0} ⊂ Δ_{0} ⊂ \overline{M}_{22}. Then we have the following equality of 2-cycles in 𝔊̃_{25}^{6}:

σ^{∗}(C^{0}) = Y′/Γ̃_{1} ≅ Γ̃_{2},

that is, σ^{∗}(C^{0}) can be naturally identified with the surface obtained from Y′ by identifying the disjoint curves Γ̃_{1} and Γ̃_{2} over each pair (y, A) ∈ C × W_{24}^{6}(C).

Proof. We fix a point y ∈ C − \{q\}, denote by [C_{y}^{0} := C/y ∼ q] ∈ \overline{M}_{22}, ν : C → C_{0}^{y}

the normalization map, and we investigate the variety \overline{W} _{25}^{6}(C_{y}^{0}) ⊂ \overline{Pic}^{25}(C_{y}^{0}) of torsionfree sheaves L on C_{y}^{0} with deg(L) = 25 and h^{0}(C_{y}^{0}, L) ≥ 7. If L ∈ W_{25}^{6}(C_{y}^{0}), that is, L is locally free, then L is determined by ν^{∗}(L) ∈ W_{25}^{6}(C) which has the property that h^{0}(C, ν^{∗}L ⊗ O_{C}(−y − q)) = 6. However, the line bundles of type A ⊗ O_{C}(y) or A ⊗ O_{C}(q) with A ∈ W_{24}^{6}(C), do not appear in this association even though they have this property. In fact they correspond to the situation when L ∈ \overline{W} _{25}^{6}(C_{0}^{y}) is not locally free, in which case necessarily L = ν_{∗}(A) for some A ∈ W_{24}^{6}(C). Thus Y ∩ π_{1}^{−1}(y) is the partial normalization of \overline{W} _{25}^{6}(C_{y}^{0}) at the n_{0} points of the form ν_{∗}(A) with A ∈ W_{24}^{6}(C). A special analysis is required when y = q, that is, when C_{y}^{0} degenerates to C ∪_{q} E_{∞}, where E_{∞} is a rational nodal cubic. If {l_{C}, l_{E_{∞}}} ∈ σ^{−1}([C ∪_{q} E_{∞}]), then an analysis along the lines of Theorem 2.1 shows that ρ(l_{C}, q) ≥ 0 and ρ(l_{E_{∞}}, q) ≤ 1. Then either l_{C} has a base point at q and then the underlying line bundle of l_{C} is of type A ⊗ O_{C}(q) while l_{E_{∞}}(−18q) ∈ \overline{W} _{7}^{6}(E_{∞}), or else, a^{l_{C}}(q) = (0, 2, 3, 4, 5, 6, 8) and then l_{E_{∞}}(−17q) ∈ P(H^{0}(E_{∞}(8q))/H^{0}(E_{∞}(6q))) ≅ E_{B}, where B ∈ W_{25}^{6}(C) is the underlying line bundle of l_{C}. □

Throughout the paper we use a few facts about intersection theory on Jacobians which we briefly recall (see [ACGH] for a general reference). If [C] ∈ M_{g} is Brill-Noether general, we denote by P the 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})).

We have the formula c_{1}(P) = dη + γ, corresponding to the Hodge decomposition of c_{1}(P). We also record that γ^{3} = γη = 0, η^{2} = 0 and γ^{2} = −2ηπ_{2}^{∗}(θ). On W_{d}^{r}(C) we have the tautological rank r + 1 vector bundle M := (π_{2})_{∗}(P_{|C×W_{d}^{r}(C)}). The Chern numbers of M can be computed using the Harris-Tu formula (cf. [HT]): if we write ∑_{i=0}^{r} c_{i}(M^{∨}) = (1 + x_{1}) ⋯ (1 + x_{r+1}), then for every class ζ ∈ H^{∗}(Pic^{d}(C), ℤ) one has the following formula:

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.

If we use the expression of the Vandermonde determinant, we get the identity

det(\frac{1}{(a_{j} + l − 1)!})_{1≤j,l≤r+1} = \frac{Π_{j>l} (a_{l} − a_{j})}{Π_{j=1}^{r+1} (a_{j} + r)!}.

By repeatedly applying this we get all intersection numbers on W_{d}^{r}(C) which we’ll need:

Lemma 2.4. If [C] ∈ M_{21} is Brill-Noether general and c_{i} := c_{i}(M^{∨}) are the Chern classes of the dual of the tautological bundle on W_{17}^{2}(C), we have the following identities in H^{∗}(W_{17}^{2}(C), ℤ):

[W_{17}^{2}(C)] = \frac{θ^{18}}{73156608000}.

x_{1} · ξ = \frac{θ^{19} · ξ}{219469824000}, x_{2} · ξ = x_{3} · ξ = 0, for anyξ ∈ H^{4}(Pic^{21}(C)).

x_{1}x_{2} · ξ = \frac{θ^{20}}{1755758592000} · ξ, x_{1}x_{3} · ξ = x_{2}x_{3} · ξ = 0, for anyξ ∈ H^{2}(Pic^{21}(C)).

x_{1}^{2} · ξ = \frac{θ^{20}}{1097349120000} · ξ, x_{2}^{2} · ξ = −x_{1}x_{2} · ξ, x_{3}^{2} · ξ = 0, for anyξ ∈ H^{2}(Pic^{21}(C)).

x_{1}^{3} = \frac{θ^{21}}{7242504192000}, x_{2}^{3} = −\frac{t^{21}}{6584094720000}, x_{3}^{3} = x_{1}x_{2}x_{3} = \frac{θ^{21}}{36870930432000}.

x_{1}^{2}x_{2} = −x_{2}^{3}, x_{1}x_{2}^{2} = x_{1}^{2}x_{3} = x_{2}x_{3}^{2} = 0, x_{1}x_{3}^{2} = x_{2}^{2}x_{3} = −x_{1}x_{2}x_{3}.

We are going to extend the vector bundles E and F over the space 𝔊̃_{25}^{6} of limit linear series:

Proposition 2.5. There exist two vector bundles E and F defined over 𝔊̃_{25}^{6} with rank(E) = 7 and rank(F) = 29 together with a vector bundle morphism Sym^{2}(E) → F, such that the following statements hold:

• For (C, L) ∈ 𝔊_{25}^{6} we have that E(L) = H^{0}(C, L) and F(L) = H^{0}(C, L^{⊗2}).

• For t = (C ∪_{y} E, l_{C}, l_{E}) ∈ σ^{−1}(Δ_{1}^{0}), where g(C) = 21, g(E) = 1 and l_{C} = |L_{C}| is such that L_{C} ∈ W_{25}^{6}(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_{y}^{0} := C/y ∼ q, L) ∈ σ^{−1}(Δ_{0}^{0}), with q, y ∈ C and L ∈ \overline{W} _{25}^{6}(C_{y}^{0}) such that h^{0}(C, ν^{∗}L ⊗ O_{C}(−y − q)) = 6, where ν : C → C_{y}^{0} 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} _{25}^{6}(C_{0}^{y})−W_{25}^{6}(C_{0}^{y}), then L = ν_{∗}(A), where A ∈ W_{24}^{6}(C) and the image of the natural map F(t) → H^{0}(C, ν^{∗}L^{⊗2}) is the subspace H^{0}(C, A^{⊗2}).

We determine the cohomology classes of the surfaces X and Y introduced in Propositions 2.3 and 2.1.

Proposition 2.6. Let [C] ∈ M_{21} be a Brill-Noether general curve and q ∈ C a general point. If M denotes the tautological rank 3 vector bundle over W_{17}^{2}(C) and c_{i} := c_{i}(M^{∨}), then one has the following relations:

(1) [X] = π_{2}^{∗}(c_{2}) − 6ηθ + (74η + 2γ)π_{2}^{∗}(c_{1}) ∈ H^{4}(C × W_{17}^{2}(C)).

(2) [Y] = π_{2}^{∗}(c_{2}) − 2ηθ + (16η + γ)π_{2}^{∗}(c_{1}) ∈ H^{4}(C × W_{17}^{2}(C)).

Proof. By Riemann-Roch, if (y, L) ∈ X, then the line bundle M := K_{C} ⊗ L^{∨} ⊗ O_{C}(2y) ∈ W_{17}^{2}(C) has a cusp at y. We realize X as the degeneracy locus of a vector bundle map over C × W_{17}^{2}(C). For each pair (y, M) ∈ C × W_{17}^{2}(C), there is a natural map

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

which globalizes to a vector bundle morphism ζ : J_{1}(P)^{∨} → π_{2}^{∗}(M)^{∨} over C × W_{17}^{2}(C) (Note that W_{17}^{2}(C) is a smooth 3-fold). Then we have the identification X = Z_{1}(ζ) and the Thom-Porteous formula gives that [X] = c_{2}(π_{2}^{∗}(M)−J_{1}(P^{∨})). From the usual exact sequence over C × Pic^{17}(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)^{∨}) = (∑_{j≥0}(17η + γ)^{j}) · (∑_{j≥0}(57η + γ)^{j}) = 1 − 6ηθ + 74η + 2γ,

which quickly leads to the formula for [X]. To compute [Y] we proceed in a similar way. We denote by p_{1}, p_{2} : C × C × Pic^{17}(C) → C × Pic^{17}(C) the two projections, by Δ ⊂ C × C × Pic^{17}(C) the diagonal and we set Γ_{q} := {q} × Pic^{17}(C). We introduce the rank 2 vector bundle B := (p_{1})_{∗}(p_{2}^{∗}(P) ⊗ O_{Δ+p_{2}^{∗}(Γ_{q})}) defined over C × W_{17}^{2}(C) and 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 + (17η + γ) + (17η + γ)^{2} + ⋯)(1 − η),

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

For future reference we also record the following formulas:

(1)  c_{3}(π_{2}^{∗}(M)^{∨} − J_{1}(P)^{∨}) = π_{2}^{∗}(c_{3}) − 6ηθπ_{2}^{∗}(c_{1}) + (74η + 2γ)π_{2}^{∗}(c_{2}) and
(2)  c_{4}(π_{2}^{∗}(M)^{∨} − J_{1}(P)^{∨}) = π_{2}^{∗}(c_{3})(74η + 2γ) − 6π_{2}^{∗}(c_{2})ηθ.

Proposition 2.7. Let [C] ∈ M_{21} be a Brill-Noether general curve and denote by P the Poincaré bundle on C × Pic^{17}(C). We have the following identities in H^{∗}(Pic^{17}(C)):

c_{1}(R^{1}π_{2∗}(P_{|C×W_{17}^{2}(C)})) = θ − c_{1} and c_{2}(R^{1}π_{2∗}(P_{|C×W_{17}^{2}(C)})) = \frac{θ^{2}}{2} − θc_{1} + c_{2}.

Proof. We recall that in order to obtain a determinantal structure on W_{17}^{2}(C) one fixes a divisor D ∈ C_{e} of degree e >> 0 and considers the morphism

(π_{2})_{∗}(P ⊗ O(π_{1}^{∗}D)) → (π_{2})_{∗}(P ⊗ O(π_{1}^{∗}D_{|π_{1}^{∗}D})).

Then W_{17}^{2}(C) is the degeneration locus of rank e − 6 of this map and there is an exact sequence of vector bundles over W_{17}^{2}(C):

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_{17}^{2}(C)}) ⟶ 0,

from which our claim easily follow once we take into account that (π_{2})_{∗}(P⊗O(π_{1}^{∗}D)_{|π_{1}^{∗}D}) is numerically trivial and c_{t}((π_{2})_{∗}(P ⊗ O(π_{1}^{∗}D))) = e^{−θ}. □

Proposition 2.8. Let [C] ∈ M_{21} and denote by p_{1}, p_{2} : C × C × Pic^{17}(C) → C × Pic^{17}(C) the natural projections. We denote by A_{2} the vector bundle on C × Pic^{17}(C) with fibre at each point A_{2}(y, M) = H^{0}(C, K_{C}^{⊗2} ⊗ M^{⊗(−2)} ⊗ O_{C}(2y)). We have the following formulas:

c_{1}(A_{2}) = −4θ − 4γ − 28η andc_{2}(A_{2}) = 8θ^{2} + 104ηθ + 16γθ.

Proof. Recall that if M ∈ W_{17}^{2}(C) then by duality L := K_{C} ⊗ O_{C}(2y) ⊗ M^{∨} ∈ W_{25}^{6}(C) is a linear series with a cusp at y. In this notation, A_{2} is the vector bundle with fibre A_{2}(y, M) = H^{0}(C, L^{⊗2} ⊗ O_{C}(−2y)). To compute c_{i}(A_{2}) we apply Grothendieck-Riemann-Roch to the map p_{2}. If ν_{1} : C × C × Pic^{17}(C) → C denotes the projection onto the first coordinate, then one obtains that

ch(p_{2!}(ν_{1}^{∗}(K_{C}^{⊗2}) ⊗ P^{⊗(−2)} ⊗ O(2Δ))) = ch(A_{2}) =

= (p_{2})_{∗}(ch(ν_{1}^{∗}(K_{C}^{⊗2}) ⊗ p_{1}^{∗}(P^{⊗(−2)}) ⊗ O(2Δ)) · (1 − \frac{1}{2}ν_{1}^{∗}(K_{C})))

and looking at terms of degree 2 and 3 one finds c_{1}(A_{2}) = −4θ−4γ −28η and ch_{2}(A_{2}) = −8ηθ. □

The next proposition is proved along the lines of Proposition 2.8:

Proposition 2.9. Let [C, q] ∈ M_{21,1} be a general pointed curve an we denote by B_{2} the vector bundle on C ×Pic^{17}(C) having fibre B_{2}(y, M) = H^{0}(C, K_{C}^{⊗2} ⊗M^{⊗(−2)} ⊗O_{C}(y +q)) at each point (y, M) ∈ C × Pic^{17}(C). Then we have that:

c_{1}(B_{2}) = −4θ + 7η − 2γ andc_{2}(B_{2}) = 8θ^{2} − 28ηθ + 8θγ.

Now we prove that the virtual degeneracy locus \overline{𝔇}_{22} is an “honest” divisor on M_{22}, that is, the vector bundle morphism Sym^{2}(E) → F is generically non-degenerate:

Theorem 2.10. If [C] ∈ M_{22} is a sufficiently general smooth curve, then the multiplication map Sym^{2}H^{0}(L) → H^{0}(L^{⊗2}) is injective for every L ∈ W_{25}^{6}(C). It follows that 𝔇_{22} is a divisor on M_{22}.

Proof. Throughout this proof we use the set-up described in Section 4 of [FP] for understanding degenerations of multiplication maps on curves. We consider a degenerate curve [S := C ∪_{q} D] ∈ Δ_{8} ⊂ \overline{M}_{22} where (C, q) and (D, q) are suitable general pointed curves of genus 8 and 14 respectively. Suppose by contradiction that [S] ∈ \overline{𝔇}_{22}. Then there exists a limit 𝔤_{25}^{6} on S, say

l = {l_{C} = (L_{C}, V_{C} ⊂ H^{0}(L_{C})), l_{D} = (L_{D}, V_{D} ⊂ H^{0}(L_{D})} ∈ σ^{−1}([S]),

together with ≠ 0 elements

ρ_{C} ∈ Ker{Sym^{2}(V_{C}) → H^{0}(L_{C}^{⊗2})} and ρ_{D} ∈ Ker{Sym^{2}(V_{D}) → H^{0}(L_{D}^{⊗2})},

such that ord_{q}(ρ_{C}) + ord_{q}(ρ_{D}) ≥ 50 = deg(L_{C}) + deg(L_{D}). Limit linear 𝔤_{25}^{6}’s on S are indexed by partitions 0 ≤ β_{0} ≤ β_{1} ≤ … ≤ β_{6} ≤ 3 such that ∑_{i=0}^{6} β_{i} = g(C) = 8. If we pick such a partition, then l_{C}(−11q) ∈ G_{14}^{6}(C) is a linear series with ramification sequence (β_{0}, …, β_{6}) at q, while l_{D}(−5q) ∈ G_{20}^{6}(D) is a linear series with complementary ramification sequence (3 − β_{6}, …, 3 − β_{0}) at q. We claim that by analyzing all the partitions (β_{i})_{0≤i≤6} one can always choose (C, q) and (D, q) general enough such that either ρ_{C} or ρ_{D} must be 0. For simplicity, we carry this out only in a single case, the other being rather similar. Say we choose the partition (β_{i})_{0≤i≤6} = (0, 0, 1, 1, 1, 2, 3). Then if L_{C} ∈ Pic^{14}(C) denotes the underlying line bundle of l_{C}(−11q), we have that a^{L_{C}}(q) = (0, 1, 3, 4, 5, 7, 9), whereas if L_{D} ∈ Pic^{20}(D) denotes the underlying line bundle of l_{D}(−5q) then a^{L_{D}}(q) = (0, 2, 4, 5, 6, 8, 9). A careful analysis similar to the one in the proof of Theorem 5.1 in [FP], shows that the only possible situation when ρ_{C} and ρ_{D} could satisfy the compatibility condition at q, is when ord_{q}(ρ_{C}) = 9(= 0+9 = 4+5). More precisely, if W_{C} ⊂ Sym^{2}H^{0}(L_{C}) denotes the 13-dimensional space of sections with order ≥ 13 at q, then

0 ≠ ρ_{C} ∈ Ker{W_{C} → H^{0}(L_{C}^{⊗2} ⊗ O_{C}(−9q))}

(Note that the kernel is indeed 1-dimensional if [C, q] ∈ M_{8,1} is sufficiently generic). By compatibility, then ord_{q}(ρ_{D}) ≥ 9(= 0 + 9 = 4 + 5). If W_{D} ⊂ Sym^{2}H^{0}(L_{D}) is the space of those ρ_{D} with ord_{q}(ρ_{D}) ≥ 9, then dim(W_{D}) = 17 and we have that

0 ≠ ρ_{D} ∈ Ker{W_{D} → H^{0}(L_{D}^{⊗2} ⊗ O_{D}(−9q))},

where the target is 18-dimensional. One has to show that this is indeed a divisorial condition on M_{14,1} , which can be seen by further degenerating [D, q] to reducible curves of smaller genus.

□

As a first step towards computing [\overline{𝔇}_{22}] we determine the δ_{1} coefficient in its expression:

Theorem 2.11. Let [C] ∈ M_{21} be Brill-Noether general and denote by C^{1} ⊂ Δ_{1} the associated test curve. Then σ^{∗}(C^{1}) · c_{2}(F − Sym^{2}(E)) = 4847375988. It follows that the coefficient of δ_{1} in the expansion of \overline{𝔇}_{22} is equal to b_{1} = 731180268.

Proof. We intersect the degeneracy locus of the map Sym^{2}(E) → F with the surface σ^{∗}(C^{1}) and use that the vector bundles E and F were defined by retaining the sections of the genus 21 aspect of each limit linear series and dropping the information coming from the elliptic curve. It follows that Z_{i} · c_{2}(F − Sym^{2}(E)) = 0 for 1 ≤ i ≤ 3 (since F and Sym^{2}(E)) are both trivial along the surfaces Z_{i}), and [X_{1}×X_{2}]·c_{2}(F −Sym^{2}(E)) = 0 (because c_{2}(F−Sym^{2}(E))_{|X_{1}×X_{2}} is in fact the pull-back of a codimension 2 class from the 1-dimensional cycle X_{1}, therefore the intersection number is 0 for dimensional reasons). We are left with estimating the contribution coming from X and we write

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

and we are going to compute each term in the right-hand-side of this expression.

Recall that we have constructed in Proposition 2.6 a vector bundle morphism ζ : J_{1}(P)^{∨} → π_{2}^{∗}(M)^{∨} and we denote by U := Ker(ζ). In other words, U is a line bundle on X with fibre

U(y, M) = \frac{H^{1}(C, M ⊗ O_{C}(−2y))^{∨}}{H^{1}(C, M)^{∨}} = \frac{H^{0}(C, L)}{H^{0}(C, L ⊗ O_{C}(−2y))}

over a point (y, M) ∈ X. The Chern class of U can be computed from the Harris-Tu formula and we find that (cf. (1)):

c_{1}(U)·ξ_{|X} = −c_{3}(π_{2}^{∗}(M)^{∨}−J_{1}(P)^{∨})·ξ_{|X} = −(π_{2}^{∗}(c_{3})−6ηθπ_{2}^{∗}(c_{1})+(74η+2γ)π_{2}^{∗}(c_{2}))·ξ_{|X},

for any class ξ ∈ H^{2}(C × W_{17}^{2}(C)), and

c_{1}^{2}(U) = c_{4}(π_{2}^{∗}(M)^{∨} − J_{1}(P)^{∨}) = π_{2}^{∗}(c_{3})(74η + 2γ) − 6π_{2}^{∗}(c_{2})ηθ.

If A_{3} denotes the rank 30 vector bundle on X having fibres

A_{3}(y, M) = H^{0}(C, L^{⊗2}) = H^{0}(C, K_{C}^{⊗2} ⊗ M^{⊗(−2)} ⊗ O_{C}(4y)),

then there is an injective bundle morphism U^{⊗2} ↪ A_{3}/A_{2} and we consider the quotient sheaf

G := \frac{A_{3}/A_{2}}{U^{⊗2}}

We note that since the morphism U^{⊗2} → A_{3}/A_{2} vanishes along the curve Γ_{0} corresponding to pairs (y, M) where M has a base point, G has torsion along Γ_{0}. 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 generic point of X corresponds to the decomposition

F(y, M) = 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 Γ_{0} as well). Hence c_{1}(F_{|X}) = c_{1}(A_{2|X}) + 2c_{1}(U) and c_{2}(F_{|X}) = c_{2}(A_{2|X}). Furthermore, we note that the vector bundle π_{2}^{∗}(R^{1}π_{2∗}(P))_{|X}^{∨} is a subbundle of E_{|X} and we have an exact sequence

0 ⟶ π_{2}^{∗}(R^{1}π_{2∗}(P))_{|X}^{∨} ⟶ E_{|X} ⟶ U ⟶ 0

from which we find that c_{1}(E_{|X}) = −θ + π_{2}^{∗}(c_{1}) + c_{1}(U). Similarly, we have that

(3)  c_{2}(E_{|X}) = \frac{θ^{2}}{2} + π_{2}^{∗}(c_{2}) − θπ_{2}^{∗}(c_{1}) − c_{1}(U)π_{2}^{∗}(c_{1}) − θc_{1}(U).

It is elementary to check that c_{1}(Sym^{2}E_{|X}) = 8 c_{1}(E_{|X}) and that c_{2}(Sym^{2}E_{|X}) = 27 c_{1}^{2}(E_{|X})+ 9 c_{2}(E_{|X}), therefore we obtain that

σ^{∗}(C^{1}) · c_{2}(F − Sym^{2}(E)) = c_{2}(A_{2|X}) + c_{1}(A_{2|X})c_{1}(U^{⊗2})−
−8c_{1}(A_{2|X})c_{1}(E_{|X}) − 8c_{1}(E_{|X})c_{1}(U^{⊗2}) + 37c_{1}^{2}(E_{|X}) − 9c_{2}(E_{|X}) =

= (−120 ηθ + \frac{17}{2}θ^{2} − 16 θγ − 9 π_{2}^{∗}(c_{2}) + (224 η + 32 γ − 33 θ)π_{2}^{∗}(c_{1}) + 37π_{2}^{∗}(c_{1}^{2})) · [X]+

+(168 η + 24 γ − 25 θ + 49 π_{2}^{∗}(c_{1})) · c_{1}(U) + 21c_{1}^{2}(U) =

= 1754 ηθπ_{2}^{∗}(c_{2}) + 1386 ηπ_{2}^{∗}(c_{3}) − 2498 ηθπ_{2}^{∗}(c_{1}^{2}) + 741 ηθ^{2}π_{2}^{∗}(c_{1}) − 4068 ηπ_{2}^{∗}(c_{1})π_{2}^{∗}(c_{2})−

−51 ηθ^{3} + 2738 ηπ_{2}^{∗}(c_{1}^{3}),

where the last expression lives inside H^{4}(C × W_{17}^{2}(C)). Using Lemma 2.4 we can evaluate each term in this sum to find that σ^{∗}(C^{1})·c_{2}(F −Sym^{2}(E)) = 691 θ^{21}/1207084032000, which implies the stated formula for b_{1}. □

Theorem 2.12. Let [C, q] ∈ M_{21,1} be a suitably general pointed curve and L ∈ W_{25}^{6}(C) a linear series with a cusp at q. Then the multiplication map Sym^{2}H^{0}(C, L) → H^{0}(C, L^{⊗2}) is injective. It follows that we have the relation a − 12b_{0} + b_{1} = 0.

Proof. We consider the pencil R ⊂ \overline{M}_{g} obtained by attaching to C at the point q a pencil of plane cubics. It is well-known that R · λ = 1, R · δ_{0} = 12 and R · δ_{1} = −1, thus the relation a − 12b_{0} + b_{1} = 0 would be immediate once we show that R ∩ \overline{𝔇}_{22} = ∅. Assume by contradiction that R ∩ \overline{𝔇}_{22} ≠ ∅ and then according to Proposition 2.1 there exists M_{C} ∈ W_{25}^{6}(C) with h^{0}(M_{C} ⊗ O_{C}(−2q)) = 6 such that the multiplication map Sym^{2}H^{0}(M_{C}) → H^{0}(M_{C}^{⊗2}) is not injective. There are two cases to consider. If q ∈ Bs|M_{C}|, then N := M_{C} ⊗ O_{C}(−q) ∈ W_{24}^{6}(C) is such that the map Sym^{2}H^{0}(N) → H^{0}(N^{⊗2}) is not an isomorphism. This is a divisorial condition on [C] ∈ M_{21} (cf. [Kh] or [F3]) and therefore it does not occur if we choose [C] ∈ M_{21} sufficiently generically. We are left with the case when M_{C} has a cusp at q, hence (q, M_{C}) ∈ X. This case is covered by Theorem 2.10 which finishes the proof.

□

Theorem 2.13. Let [C, q] ∈ M_{21,1} be a Brill-Noether general pointed curve and denote by C^{0} ⊂ Δ_{0} the associated test curve. Then σ^{∗}(C^{0})·c_{2}(F −Sym^{2}(E)) = 42b_{0}−b_{1} = 4847375988. It follows that b_{0} = 132822768.

Proof. This time we look at the virtual degeneracy locus of the morphism Sym^{2}(E) → F along the surface σ^{∗}(C^{0}). The first thing to note is that the vector bundles E_{|σ^{∗}(C^{0})} and F_{|σ^{∗}(C^{0})} are both pull-backs of vector bundles on Y. For convenience we denote this vector bundles also by E and F, hence to use the notation of Proposition 2.3, E_{|σ^{∗}(C^{0})}) = ϵ^{∗}(E_{|Y}) and F_{|σ^{∗}(C^{0})} = ϵ^{∗}(F_{|Y}). We find that

σ^{∗}(C^{0}) · c_{2}(F − Sym^{2}(E)) = c_{2}(F_{|Y}) − c_{1}(F_{|Y}) · c_{1}(E_{|Y}) + c_{1}^{2}(E_{|Y}) − c_{2}(E_{|Y})

and like in the proof of Theorem 2.11, we are going to compute each term in this expression. We denote by V := Ker(χ), where χ : B^{∨} → π_{2}^{∗}(M)^{∨} is the bundle morphism coming from Proposition 2.6. Thus V is a line bundle on Y with fibre

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

over each point (y, M) ∈ Y, and where L := K_{C} ⊗ M^{∨} ⊗ O_{C}(y + q) ∈ W_{25}^{6}(C). By using again the Harris-Tu Theorem, we find the following formulas for the Chern numbers of V:

c_{1}(V) · ξ_{|Y} = −(c_{3}(π_{2}^{∗}(M)^{∨} − B^{∨}) · ξ_{|Y}) = (π_{2}^{∗}(c_{3}) + π_{2}^{∗}(c_{2})(16η + γ) − 2π_{2}^{∗}(c_{1})ηθ) · ξ_{|Y} ,

for any class ξ ∈ H^{2}(C × W_{17}^{2}(C)), and

c_{1}^{2}(V) = c_{4}(π_{2}^{∗}(M)^{∨} − B^{∨}) = π_{2}^{∗}(c_{3})(16η + γ) − 2π_{2}^{∗}(c_{2})ηθ.

Recall that we introduced the rank 28 vector bundle B_{2} over C × W_{17}^{2}(C) with fibre B_{2}(y, M) = H^{0}(C, L^{⊗2} ⊗ O_{C}(−y − q)). We claim that one has an exact sequence of bundles over Y

(4)  0 ⟶ B_{2|Y} ⟶ F_{|Y} ⟶ V^{⊗2} ⟶ 0.

If B_{3} is the rank 30 vector bundle on Y with fibres

B_{3}(y, M) = H^{0}(C, L^{⊗2}) = H^{0}(C, K_{C}^{⊗2} ⊗ M^{⊗(−2)} ⊗ O_{C}(2y + 2q)),

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 (4) now becomes clear. Therefore c_{1}(F_{|Y}) = c_{1}(B_{2|Y}) + 2c_{1}(V) and c_{2}(F_{|Y}) = c_{2}(B_{2|Y}). Reasoning along the lines of Theorem 2.11, we also have an exact sequence

0 ⟶ π_{2}^{∗}(R^{1}π_{2∗}(P))_{|Y}^{∨} ⟶ E_{|Y} ⟶ V ⟶ 0

and from this we obtain that

c_{1}(E_{|Y}) = −θ+π_{2}^{∗}(c_{1})+c_{1}(V) and c_{2}(E_{|Y}) = \frac{θ^{2}}{2} +π_{2}^{∗}(c_{2})−θπ_{2}^{∗}(c_{1})−θc_{1}(V)+c_{1}(V)π_{2}^{∗}(c_{1}).

All in all, we can write the following expression for the total intersection number

σ^{∗}(C^{0}) · c_{2}(F − Sym^{2}(E)) = c_{2}(B_{2|Y}) + c_{1}(B_{2|Y})c_{1}(V^{⊗2})−
−8c_{1}(B_{2|Y})c_{1}(E_{|Y}) − 8c_{1}(E_{|Y})c_{1}(V^{⊗2}) + 37c_{1}^{2}(E_{|Y}) − 9c_{2}(E_{|Y}) =

= (\frac{17}{2}θ^{2} + 28ηθ − 8θγ − 9π_{2}^{∗}(c_{2}) + (16γ − 33θ − 56η)π_{2}^{∗}(c_{1}) + 37π_{2}^{∗}(c_{1}^{2})) · [Y]+

+(49 π_{2}^{∗}(c_{1}) − 25 θ − 42 η + 12 γ)c_{1}(V) + 21c_{1}^{2}(V) =
= 428 ηθπ_{2}^{∗}(c_{2}) − 536 ηθπ_{2}^{∗}(c_{1}^{2}) + 168 ηθ^{2}π_{2}^{∗}(c_{1}) − 984 ηπ_{2}^{∗}(c_{1})π_{2}^{∗}(c_{2})+
+378ηπ_{2}^{∗}(c_{3}) − 17 ηθ^{3} + 592ηπ_{2}^{∗}(c_{1}^{3}),

and using once more Lemma 2.4, we get that 42b_{0} − b_{1} = 509θ^{21}/5364817920000. Since we already know the value of b_{1} and a − 12b_{0} + b_{1} = 0, this allows us to calculate a and b_{0}. □

End of the proof of Theorem 1.1. We write \overline{𝔇}_{22} ≡ aλ − ∑_{j=0}^{11} b_{j}δ_{j}. Since a/b_{0} = 17121/2636 ≤ 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 ≤ 11, hence s(\overline{𝔇}_{22}) = a/b_{0} < 13/2. □

References

[AC1] E. Arbarello and M. Cornalba, Footnotes to a paper of Beniamino Segre, Math. Ann. 256 (1981), 341-362.

[AC2] E. Arbarello and M. Cornalba, A few remarks about the variety of irreducible plane curves of given degree and genus, Ann. Scient. Éc. Norm. Sup. 16 (1983), 467-488.

[ACGH] E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of algebraic curves, Grundlehren der mathematischen Wissenschaften 267, Springer.

[CR1] M.C. Chang and Z. Ran, Unirationality of the moduli space of curves of genus 11, 13 (and 12), Invent. Math. 76 (1984), 41-54.

[CR2] M.C. Chang and Z. Ran, The Kodaira dimension of the moduli space of curves of genus 15, J. Diff. Geom. 24 (1986), 205-220.

[EH] D. Eisenbud and J. Harris, The Kodaira dimension of the moduli space of curves of genus ≥ 23, Invent. Math. 90 (1987), 359-387.

[H1] J. Harris, On the Kodaira dimension of the moduli space of curves II: The even genus case, Invent. Math. 75 (1984), 437-466.

[H2] J. Harris, On the Severi problem, Invent. Math. 84 (1984), 445-461.

[HM] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23-88.

[HMo] J. Harris and I. Morrison, Slopes of effective divisors on the moduli space of stable curves, Invent. Math. 99 (1990), 321-355.

[HT] J. Harris and L. Tu, Chern numbers of kernel and cokernel bundles, Invent. Math. 75 (1984), 467-475.

[F1] G. Farkas, The geometry of the moduli space of curves of genus 23, Math. Ann. 318 (2000), 43-65.

[F2] G. Farkas, Syzygies of curves and the effective cone of \overline{M}_{g}, math.AG/0503498, to appear in Duke Math. J. (2006).

[F3] G. Farkas, Koszul divisors on moduli spaces of curves, math.AG/0607475.

[FP] G. Farkas and M. Popa, Effective divisors on \overline{M}_{g}, curves on K3 surfaces and the Slope Conjecture, J. Algebraic Geometry, 14 (2005), 241-267.

[Kh] D. Khosla, Moduli spaces of curves with linear series and the Slope Conjecture, math.AG/0608024.

[Se] E. Sernesi, L’unirazionalita della varieta dei moduli delle curve di genere dodici, Ann. Scuola Norm. Sup. Pisa 8 (1981), 405-439.

[V] A. Verra, The unirationality of the moduli space of curves of genus 14 or lower, Compositio Math. 141 (2005), 1425-1444.

\overline{M}_{22} IS OF GENERAL TYPE 13

Department of Mathematics, University of Texas, Austin, TX 78712

E-mail address: gfarkas@math.utexas.edu