THE UNIVERSAL ABELIAN VARIETY OVER A_{5}

GAVRIL FARKAS AND ALESSANDRO VERRA

Abstract. We establish a structure result for the universal abelian variety over A_{5}. This implies that the boundary divisor of \overline{A}_{6} is unirational and leads to a lower bound on the slope of the cone of effective divisors on \overline{A}_{6}.

The general principally polarized abelian variety [A, Θ] ∈ A_{g} of dimension g ≤ 5 can be realized as a Prym variety. Abelian varieties of small dimension can be studied in this way via the rich and concrete theory of curves, in particular, one can establish that A_{g} is unirational in this range. In the case g = 5, the Prym map P : R_{6} → A_{5} is finite of degree 27, see [DS]; three different proofs [Don], [MM], [Ve1] of the unirationality of R_{6} are known. The moduli space A_{g} is of general type for g ≥ 7, see [Mu], [T]. Determining the Kodaira dimension of A_{6} is a notorious open problem.

The aim of this paper is to give a simple unirational parametrization of the universal abelian variety over A_{5} and hence of the boundary divisor of a compactification of A_{6}. We denote by ϕ : X_{g−1} → A_{g−1} the universal abelian variety of dimension g − 1. The moduli space Ã_{g} of principally polarized abelian varieties of dimension g and their rank 1 degenerations is a partial compactification of A_{g} obtained by blowing-up A_{g−1} in the Satake compactification, cf. [Mu]. Its boundary ∂Ã_{g} is isomorphic to the universal Kummer variety in dimension g − 1 and there exist a surjective double covering j : X_{g−1} → ∂Ã_{g}. We establish a simple structure result for the boundary ∂Ã_{6}:

Theorem 0.1. The universal abelian variety X_{5} is unirational.

This immediately implies that the boundary divisor ∂Ã_{6} is unirational as well. What we prove is actually stronger than Theorem 0.1. Over the moduli space R_{g} of smooth Prym curves of genus g, we consider the universal Prym variety φ : Y_{g} → R_{g} obtained by pulling-back X_{g−1} → A_{g−1} via the Prym map P : R_{g} → A_{g−1}. Let \overline{R}_{g} be the moduli space of stable Prym curves of genus g together with the universal Prym curve π̃ : C̃ → \overline{R}_{g} of genus 2g − 1. If C̃^{g−1} := C̃×_{\overline{R}_{g}} … ×_{\overline{R}_{g}} C̃is the (g − 1)-fold product,

one has a universal Abel-Prym rational map 𝔞𝔭 : C̃^{g−1} −−→ Y_{g}, whose restriction on each individual Prym variety is the usual Abel-Prym map. The rational map 𝔞𝔭 is dominant and generically finite (see Section 4 for details). We prove the following result:

Theorem 0.2. The five-fold product C̃^{5} of the universal Prym curve over \overline{R}_{6} is unirational.

The key idea in the proof of Theorem 0.2 is to view smooth Prym curves of genus 6 as discriminants of conic bundles, via their representation as symmetric determinants of quadratic forms in three variables. We fix four general points u_{1}, …, u_{4} ∈ P^{2}, set w_{i} := (u_{i}, u_{i}) ∈ P^{2} × P^{2}, then consider the linear system

P^{15} := |I_{{w_{1},...,w_{4}}}^{2}(2, 2)| ⊂ |O_{P^{2}×P^{2}}(2, 2)|

of hypersurfaces Q ⊂ P^{2} ×P^{2} of bidegree (2, 2) which are nodal at w_{1}, …, w_{4}. Fixing the points w_{1}, …, w_{4} ∈ P^{2} × P^{2} neutralizes the action of Aut(P^{2}) × Aut(P^{2}). For a general threefold Q ∈ P^{15}, the first projection p : Q → P^{2} induces a conic bundle structure with a sextic discriminant curve Γ ⊂ P^{2} such that p(Sing(Q)) = Sing(Γ). The discriminant curve Γ is nodal precisely at the points u_{1}, …, u_{4}. Furthermore, Γ is equipped with an unramified double cover p_{Γ} : Γ̃ → Γ, parametrizing the lines which are components of the singular fibres of p : Q → P^{2}. By normalizing, p_{Γ} lifts to an unramified double cover f : C̃ → C between the normalization C̃ of Γ̃ and the normalization C of Γ respectively. Note that there exists an exact sequence of generalized Prym varieties

0 ⟶ (C^{∗})^{4} ⟶ P (Γ̃/Γ) ⟶ P (C̃/C) ⟶ 0.

One can show without much effort that the assignment P^{15} ∋ Q ↦ [C̃ →^{f} C] ∈ R_{6} is dominant. This offers an alternative, much simpler, proof of the unirationality of R_{6}. However, much more can be obtained with this construction.

Let G := P^{2} × (P^{2})^{∨} = {(o, ℓ) : o ∈ P^{2}, ℓ ∈ {o} × (P^{2})^{∨}} be the Hilbert scheme of lines in the fibres of the first projection p : P^{2} × P^{2} → P^{2}. Since containing a given line in a fibre of p imposes three conditions on the linear system P^{15} of threefolds Q ⊂ P^{2}×P^{2} as above, it follows that imposing the condition {o_{i}} × ℓ_{i} ⊂ Q for five general lines, singles out a unique conic bundle Q ∈ P^{15}. This induces an étale double cover f : C̃ → C, as above, over a smooth curve of genus 6. Moreover, f comes equipped with five marked points ℓ_{1}, …, ℓ_{5} ∈ C̃. To summarize, we have a well-defined rational map

ζ : G^{5} −−→ C̃^{5}, ζ((o_{1}, ℓ_{1}), …, (o_{5}, ℓ_{5})) := (f : C̃ → C, ℓ_{1}, …, ℓ_{5}),

between two 20-dimensional varieties.

Theorem 0.3. The morphism ζ : G^{5} −−→ C̃^{5} is dominant, so that C̃^{5} is unirational.

More precisely, we show that G^{5} is birationally isomorphic to the fibre product P^{15} ×_{\overline{R}_{6}} C̃^{5}. In order to set Theorem 0.3 on the right footing and in view of enumerative calculations, we introduce a P^{2}-bundle π : P(M) → S over the quintic del Pezzo surface S obtained by blowing-up P^{2} at the points u_{1}, …, u_{4}. The rank 3 vector bundle M on S is obtained by making an elementary transformation along the four exceptional divisors E_{1}, …, E_{4} over u_{1}, …, u_{4}. The nodal threefolds Q ⊂ P^{2} × P^{2} considered above can be thought of as sections of a tautological linear system on P(M), and via the identification

|I_{{w_{1},...,w_{4}}}^{2}(2, 2)| = |O_{P(M)}(2)|,

we can view 4-nodal conic bundles in P^{2} × P^{2} as smooth conic bundles over S. In this way the numerical characters of a pencil of such conic bundles can be computed (see Sections 2 and 3 for details).

Theorem 0.3 is then used to give a lower bound for the slope of the effective cone of \overline{A}_{6} (though we stop short of determining the Kodaira dimension of \overline{A}_{6}). Recall that if E is an effective divisor on the perfect cone compactification \overline{A}_{g} of A_{g} with no component supported on the boundary D_{g} := \overline{A}_{g} − A_{g} and [E] = aλ_{1} − b[D_{g}], where λ_{1} ∈ CH^{1}(Ã_{g}) is the Hodge class, then the slope of E is defined as s(E) := \frac{a}{b} ≥ 0. The slope s(\overline{A}_{g}) of the effective cone of divisors of \overline{A}_{g} is the infimum of the slopes of all effective divisors on \overline{A}_{g}. This important invariant governs to a large extent the birational geometry of A_{g}; for instance, since K_{\overline{A}_{g}} = (g + 1)λ_{1} − [∂\overline{A}_{g}], the variety \overline{A}_{g} is of general type if s(\overline{A}_{g}) < g + 1, and uniruled when s(\overline{A}_{g}) > g + 1.

It is known that s(\overline{A}_{4}) = 8 and that the Jacobian locus \overline{M}_{4} ⊂ \overline{A}_{4} achieves the minimal slope [SM]; one of the results of [FGSMV] is the calculation s(\overline{A}_{5}) = \frac{54}{7}. Furthermore, the only irreducible effective divisor on \overline{A}_{5} of minimal slope is the closure of the Andreotti-Mayer divisor N′_{0} consisting of 5-dimensional ppav [A, Θ] for which the theta divisor Θ is singular at a point which is not 2-torsion. Concerning \overline{A}_{6}, we establish the following estimate:

Theorem 0.4. The following lower bound holds: s(\overline{A}_{6}) ≥ \frac{53}{10}.

Note that this is the first concrete lower bound on the slope of \overline{A}_{6}. The idea of proof of Theorem 0.4 is very simple. Since C̃^{5} is unirational, we choose a suitable sweeping rational curve i : P^{1} → C̃^{5}, which we then push forward to \overline{A}_{6} as follows:

\xymatrix{P^{1} \ar[r]^{i} \ar[rrrr]^{h} & C̃^{5} \ar[r]^{𝔞𝔭} & Ỹ_{6} \ar[r] & X̃_{5} \ar[r]^{j} & ∂\overline{A}_{6}}

Here Ỹ_{6} and X̃_{5} are partial compactifications of Y_{6} and X_{5} respectively, which are described in Section 4. The curve h(P^{1}) sweeps the boundary divisor of \overline{A}_{6} and intersects non-negatively any effective divisor on \overline{A}_{6} not contained in ∂\overline{A}_{6}. In particular,

s(\overline{A}_{6}) ≥ \frac{h(P^{1}) · [∂\overline{A}_{6}]}{h(P^{1}) · λ_{1}}.

To define i : P^{1} → C̃^{5}, we fix general points (o_{1}, ℓ_{1}), …, (o_{4}, ℓ_{4}) ∈ G and a further general point o ∈ P^{2}. Then we consider the image under ζ of the pencil of lines in P^{2} through o, that is, the sweeping curve i is defined as

P(T_{o}(P^{2})) ∋ ℓ ↦ ζ((o_{1}, ℓ_{1}), …, (o_{4}, ℓ_{4}), (o, ℓ)) ∈ C̃^{5}.

The calculation of the numerical characters of h(R) ⊂ \overline{A}_{6} is a consequence of the geometry of the map ζ and is completed in Section 4. Acknowledgment: We are grateful to Sam Grushevsky for explaining to us important aspects of the geometry of the universal Kummer variety (especially Proposition 4.1).

1. Determinantal nodal sextics and a parametrization of X_{5}

In this section we prove Theorem 0.3. We begin by recalling basic facts about determinantal representation of nodal plane sextics, see [B2], [Dol], [DIM]. Let Γ ⊂ P^{2} be an integral 4-nodal sextic and ν : C → Γ the normalization map, thus C is a smooth curve of genus 6. One has an exact sequence at the level of 2-torsion groups

0 ⟶ ℤ_{2}^{⊕4} ⟶ Pic^{0}(Γ)[2] ⟶^{ν^{∗}} Pic^{0}(C)[2] ⟶ 0.

In particular, unsplit étale double covers f : Γ′ → Γ are induced by 2-torsion points η ∈ Pic^{0}(Γ)[2], such that η_{C} := ν^{∗}(η) ≠ O_{C}.

Definition 1.1. We denote by P_{6} the quasi-projective moduli space of pairs (Γ, η) as above, where Γ ⊂ P^{2} is an integral 4-nodal sextic and η ∈ Pic^{0}(Γ)[2] is a torsion point inducing a non-split double cover Γ′ → Γ, or equivalently, η_{C} ≠ O_{C}.

Starting with a general element [C, η_{C}] ∈ R_{6}, since |W_{6}^{2}(C)| = 5, there are five sextic nodal plane models ν : C → Γ. For each of them, there are 2^{4} further ways of choosing η ∈ (ν^{∗})^{−1}(η_{C}). Thus there is a degree 80 = 5 · 2^{4} covering ρ : P_{6} → R_{6}.

Suppose now that (Γ, η) ∈ P_{6} is a general point^{1}. In particular h^{0}(Γ, η(1)) = 0, or equivalently, h^{0}(Γ, η(2)) = 3. Indeed, the condition h^{0}(Γ, η(1)) ≥ 1 implies that Γ ⊂ P^{2} possesses a totally tangent conic, that is, there exists a reduced conic B ⊂ P^{2} such that ν^{∗}(B) = 2b, with b being an effective divisor of C. This condition is satisfied only if ρ(Γ, η) lies in the ramification divisor of the Prym map P : R_{6} → A_{5}, see [FGSMV]. Thus we may assume that h^{0}(Γ, η(2)) = 3, for a general point (Γ, η) ∈ P_{6}.

Following [B2] Theorem B, it is known that such a sheaf η admits a resolution

(1)  0 ⟶ O_{P}_{2}(−4)^{⊕3} ⟶^{A} O_{P}_{2}(−2)^{⊕3} ⟶ η ⟶ 0,

where the map A is given by a symmetric matrix (a_{ij}(x_{1}, x_{2}, x_{3}))_{i,j=1}^{3} of quadratic

forms. More precisely, we can view the resolution (1) as a twist of the exact sequence

(2)  0 ⟶ H^{0}(Γ, η(2))^{∨} ⊗ O_{P}_{2}(−2) ⟶^{A} H^{0}(Γ, η(2)) ⊗ O_{P}_{2} ⟶^{ev} η(2) ⟶ 0,

where ev is the evaluation map. Since η is invertible, for each point x ∈ Γ one has

1 = dim_{ℂ} η(x) = 3 − rk A(x),

where, as usual, η(x) := η_{x} ⊗_{O_{Γ,x}} ℂ(x) is the fibre of the sheaf η at the point x. Thus rk A(x) = 2, for each x ∈ Γ.

To the matrix A ∈ M_{3}(H^{0}(O_{P}_{2}(2))) we can associate the following (2, 2) threefold in P_{[x_{1}:x_{2}:x_{3}]}^{2} × P_{[y_{1}:y_{2}:y_{3}]}^{2} = P^{2} × P^{2}

Q : ∑_{i,j=1}^{3} a_{ij}(x_{1}, x_{2}, x_{3})y_{i}y_{j} = 0,

which is a conic bundle with respect to the two projections. Alternatively, if

A : H^{0}(Γ, η(2))^{∨} ⊗ H^{0}(Γ, η(2))^{∨} → H^{0}(Γ, O_{Γ}(2))

is the symmetric map appearing in (2), then A induces the (2, 2) hypersurface

Q ⊂ P(H^{0}(Γ, O_{Γ}(1))^{∨}) × P(H^{0}(Γ, η(2))^{∨}) = P^{2} × P^{2}.

We denote by p : Q → P^{2} the first projection and then Γ ⊂ P^{2} is precisely the discriminant curve of Q given by determinantal equation Γ := {det A(x_{1}, x_{2}, x_{3}) = 0}. Let Γ′ denote the Fano scheme of lines F_{1}(p^{−1}(Γ)/Γ) over the discriminant curve Γ and f : Γ′ → Γ be the forgetful map f(x, ℓ) := x, where ℓ is an irreducible component of p^{−1}(x). Since rk A(x) = 2 for all x ∈ Γ, it follows that f is an étale double cover.

Proposition 1.2. For (Γ, η) ∈ P_{6}, the restriction map p_{|Sing(Q)} : Sing(Q) → Sing(Γ) is bijective.

1We shall soon establish that P_{6} is irreducible, but here we just require that ρ(Γ, η) be a general point of the irreducible variety R_{6}.

THE UNIVERSAL ABELIAN VARIETY OVER A_{5} 5

Proof. Let x ∈ Γ and R := O_{P^{2},x} be the local ring of P^{2} and 𝔪 its maximal ideal. After a linear change of coordinates, we may assume that the matrix A mod 𝔪 =: A(x) equals

A(x) = (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix})

Suppose (x, y = [y_{1}, y_{2}, y_{3}]) ∈ Sing(Q). Then A(x) ·^{t} y = 0, hence y_{1} = y_{2} = 0. Imposing that the partials of the defining equation of Q with respects to x_{1}, x_{2}, x_{3} vanish, we obtain that a_{33} ∈ 𝔪^{2}. Since det(a_{ij}) ≡ a_{33} mod 𝔪^{2}, this implies that Γ is singular at x. Conversely, for x ∈ Sing(Γ), we obtain that Sing(Q) ∩ p^{−1}(x) = {(x, y)}, where y ∈ P^{2} is uniquely determined by the condition A(x)·^{t}y = 0 (use once more that rk A(x) = 2). □ Proposition 1.3. f_{∗}(O_{Γ′}) = O_{Γ} ⊕ η, that is, the double cover f is induced by η. Proof. Essentially identical to [B1] Lemme 6.14. □

Summarizing the discussion so far, to give a general point (Γ, η) ∈ P_{6} is equivalent to specify a 4-nodal conic bundle as above. Let T ⊂ |O_{P^{2}×P^{2}}(2, 2)| be the subvariety consisting of 4-nodal hypersurfaces of bidegree (2, 2). This is an irreducible 31-dimensional variety endowed with an action of Aut(P^{2} × P^{2}). Theorem 1.4. A general Prym curve (Γ, η) ∈ P_{6} is the discriminant of a 4-nodal conic bundle p : Q → P^{2}, where Q ⊂ P^{2} × P^{2} is a 4-nodal threefold of bidegree (2, 2). More precisely, we have a birational isomorphism T//Aut(P^{2} × P^{2}) −−→^{≅} P_{6}. Remark 1.5. A similar isomorphism between the moduli space of Prym curves over smooth plane sextics and the quotient |O_{P^{2}×P^{2}}(2, 2)|//Aut(P^{2} × P^{2}) has already been established and used in [B1] and [Ve2]. Remark 1.6. Theorem 1.4 yields another (shorter) proof of the unirationality of R_{6}.

The automorphism group of P^{2} × P^{2} sits in an exact sequence

0 ⟶ P GL(3) × P GL(3) ⟶ Aut(P^{2} × P^{2}) ⟶ ℤ_{2} ⟶ 0.

In particular, we can fix four general points u_{1}, …, u_{4} ∈ P^{2}, as well as diagonal points w_{i} := (u_{i}, u_{i}) ∈ P^{2} × P^{2}, and consider the linear system P^{15} := |I_{{w_{1},...,w_{4}}}^{2}(2, 2)| of (2, 2) threefolds with assigned nodes at these points. Theorem 1.4 implies the existence of a dominant discriminant map 𝔡 : P^{15} −−→ P_{6} assigning 𝔡(Q) := (Γ′ →^{f} Γ). Proof of Theorem 0.3. Using the notation introduced in this section and in the Introduction, setting μ := φ ∘ 𝔞𝔭 : C̃_{5} −−→ \overline{R}_{6}, one has the following commutative diagram:

\xymatrix{G^{5} \ar[r] \ar[d] & P_{6} ×_{R_{6}} C̃^{5} \ar[d] \ar[r] & C̃^{5} \ar[d]_{μ} \\ _{P_{15}} \ar[r]^{𝔡} & P_{6} \ar[r]^{ρ} & _{\overline{R}_{6}}}

The dominance of the composite map ζ : G^{5} −−→ C̃^{5} follows once we observe, that the above diagram is birationally a fibre product, that is, G^{5} −−→^{≅} P^{15} ×_{\overline{R}_{6}} C̃^{5}. □

2. Conic bundles over a del Pezzo surface

With view to further applications, we analyze the linear system of conic bundles of type (2, 2) in P^{2} × P^{2} which are singular at four fixed general points and birationally, we reconstruct such a linear system as the complete linear system of smooth conic bundles in a certain P^{2}-bundle over a smooth quintic del Pezzo surface.

We fix four general points u_{1}, …, u_{4} ∈ P^{2} and set w_{i} := (u_{i}, u_{i}) ∈ P^{2} ×P^{2}. Let S be the Del Pezzo surface defined by the blow-up σ : S → P^{2} of u_{1}, …, u_{4}. For i = 1, …, 4, we denote by E_{i} := σ^{−1}(u_{i}) the exceptional line over u_{i}. Set E := E_{1} + ⋯ + E_{4} and denote by L ∈ |σ^{∗}O_{P^{2}}(1)| the pull-back of a line in P^{2} under σ. An important role is played by the rank 3 vector bundle M on S defined by the following sequence

(3)  0 ⟶ M ⟶^{j} H^{0}(S, L) ⊗ O_{S}(L) ⟶^{r} ⨁_{i=1}^{4} O_{E_{i}}(2L) ⟶ 0.

Here r_{i} : H^{0}(S, O_{S}(L)) ⊗ O_{S}(L) → O_{E_{i}}(2L) is the evaluation map and r := ⊕_{i=1}^{4}r_{i}. Since O_{E_{i}}(L) is trivial, it follows that h^{0}(r) is surjective. Passing to cohomology, we write the exact sequence

0 ⟶ H^{0}(S, M) ⟶^{h^{0}(j)} H^{0}(S, O_{S}(L)) ⊗ H^{0}(S, O_{S}(L)) ⟶^{h^{0}(r)} ⨁_{i=1}^{4} H^{0}(O_{E_{i}}(2L)) ⟶ 0.

In particular, we obtain that h^{0}(S, M) = 5. By direct calculation, we also find that

(4)  c_{1}(M) = O_{S}(−K_{S}) and c_{2}(M) = 3.

Under the decomposition H^{0}(O_{S}(L)) ⊗ H^{0}(O_{S}(L)) = ∧^{2}H^{0}(O_{S}(L)) ⊕ H^{0}(O_{S}(2L)) into symmetric and anti-symmetric tensors, the space j(H^{0}(S, M)) decomposes as

H^{0}(S, M) = H^{0}(S, M)^{−} ⊕ H^{0}(S, M)^{+} = ⋀^{2} H^{0}(S, O_{S}(L)) ⊕ H^{0}(S, O_{S}(2L − E)).

Lemma 2.1. The vector bundle M is globally generated.

Proof. Clearly, we only need to address the global generation of M along ⋃_{i=1}^{4} E_{i} and to that end, we consider the restriction of the sequence (3) to E_{i},

M_{|E_{i}} ⟶^{j_{|E_{i}}} H^{0}(S, O_{S}(L)) ⊗ O_{E_{i}} ⟶^{r_{|E_{i}}} O_{E_{i}} ⟶ 0.

The sheaf H^{0}(O_{S}(L − E_{i})) ⊗ O_{E_{i}} = O_{P^{1}}^{⊕2} is the kernel of r_{|E_{i}}. Since det(M_{|E_{i}}) = O_{E_{i}}(1), it follows that M_{|E_{i}} fits into an exact sequence of bundles on P^{1}:

0 ⟶ O_{P^{1}}(1) ⟶ M_{|E_{i}} ⟶^{j|E_{i}} O_{P^{1}}^{⊕2} ⟶ 0.

This sequence is split, so that M_{|E_{i}} = O_{P^{1}}(1) ⊕ O_{P^{1}}^{⊕2}, which is globally generated. The same holds for M if the map H^{0}(M) → H^{0}(M_{|E_{i}}) is surjective; this is implied by the vanishing H^{1}(S, M(−E_{i})) = 0. We twist by O_{S}(−E_{i}) the sequence (3), and write

0 ⟶ M(−E_{i}) ⟶ H^{0}(S, O_{S}(L)) ⊗ O_{S}(L − E_{i}) ⟶^{r} ⨁_{i=1}^{4} O_{E_{i}}(2L − E_{i}) ⟶ 0.

Since h^{0}(r) is surjective and h^{1}(S, O_{S}(L − E_{i})) = 0, it follows h^{1}(S, M(−E_{i})) = 0. □

From now on we set P := P(M) and consider the P^{2}-bundle π : P → S. The linear system |O_{P}(1)| is base point free, for M is globally generated. We reserve the notation

h := ϕ_{O_{P}(1)} : P → P^{4} := PH^{0}(S, M)^{∨}.

for the induced morphism. A Chern classes count implies that deg(h) = 2. The map j from the sequence (3) induces a birational morphism

ϵ : S × P^{2} −−→ P.

We describe a factorization of ϵ. Since j is an isomorphism along U := S − ⋃_{i=1}^{4} E_{i}, it follows that ϵ : U × P^{2} → π^{−1}(U) is biregular. The behaviour of ϵ along E_{i} × P^{2} can be understood in terms of the restriction of the sequence (3) to E_{i}. Following the proof of Lemma 2.1, one has the exact sequence

0 ⟶ O_{P}_{1}(1) ⟶ M_{|E_{i}} ⟶^{j_{|E_{i}}} H^{0}(S, O_{S}(L)) ⊗ O_{E_{i}} ⟶^{r_{|E_{i}}} O_{E_{i}} ⟶ 0,

where Im(j_{|E_{i}}) = H^{0}(O_{S}(L − E_{i})) ⊗ O_{E_{i}}. Now j_{|E_{i}} induces a rational map

ϵ_{|E_{i}×P^{2}} : E_{i} × P^{2} −−→ P(M_{|E_{i}}) ⊂ P.

For a point x ∈ E_{i}, the restriction of ϵ to P^{2} × {x} is the projection {x} × P^{2} → P^{1} of center (x, u_{i}). This implies that:

Lemma 2.2. The birational map ϵ contracts E_{i} × P^{2} to a surface which is a copy of P^{1} × P^{1}. Furthermore, the indeterminacy scheme of ϵ is equal to ⋃_{i=1}^{4} E_{i} × {u_{i}}.

Let D_{i} := E_{i} × {u_{i}} ⊂ S × P^{2} and D := D_{1} + ⋯ + D_{4}. We consider the blow-up

α : { \atop S × P^{2}} → S × P^{2}

of S × P^{2} along D, and the birational map

ϵ_{2} := ϵ ∘ α : { \atop S × P^{2}} → P.

The restriction of ϵ_{2} to the strict transform { \atop E_{i} × P^{2}} of E_{i} ×P^{2} is a regular morphism, for ϵ_{|E_{i}×P^{2}} is defined by the linear system |I_{E_{i}×{u_{i}}/S×P^{2}}(1, 1)|. This implies that ϵ_{2} itself is a regular morphism:

Proposition 2.3. The following commutative diagram solves the indeterminacy of ϵ:

\xymatrix{ & { \atop S × P^{2}} \ar[dl]_{α} \ar[dr]^{ϵ_{2}} \\ S × P^{2} \ar@{.>}[rr]^{ϵ} &  & P}

In the sequel, it will be useful to consider the exact commutative diagram

\xymatrix{H^{0}(S, M) \ar[r] \ar[d] & H^{0}(O_{S}(L)) ⊗ H^{0}(O_{S}(L)) \ar[d] \ar[r] & ⨁_{i=1}^{4} H^{0}(O_{E_{i}}(2L)) \ar[d] \\ H^{0}(I_{\{w_{1}...w_{4}\}}(1, 1)) \ar[r] & H^{0}(O_{P}_{2}(1)) ⊗ H^{0}(O_{P}_{2}(1)) \ar[r] & ⨁_{i=1}^{4} H^{0}(O_{w_{i}}(2))}

8 G. FARKAS AND A. VERRA

where the vertical arrows are isomorphisms induced by σ : S → P^{2}. Starting from the left arrow, one can construct the commutative diagram

\xymatrix{H^{0}(S, M) ⊗ O_{S} \ar[r] \ar[d] & M \ar[d]^{j} \\ H^{0}(I_{\{w_{1}...w_{4}\}}(1, 1)) ⊗ O_{S} \ar[r] & H^{0}(S, L) ⊗ O_{S}(L)}

Passing to evaluation maps, we obtain the morphism h : P → P^{4} and the rational map h_{D} : S × P^{2} −−→ P^{4} defined by the space (σ × id_{P^{2}})^{∗}H^{0}(P^{2} × P^{2}, I_{{w_{1},...,w_{4}}}(1, 1)).

The discussion above is summarized in the following commutative diagram:

\xymatrix{ & { \atop S × P^{2}} \ar[dl]_{α} \ar[dr]^{ϵ_{2}} \\ S × P^{2} \ar@{.>}[rr]^{ϵ} \ar@{.>}[dr]_{h_{D}} &  & P \ar[dl]^{h} \\  & P^{4}}

We derive a few consequences. Let π_{1} : S × P^{2} → S and π_{2} : S × P^{2} → P^{2} be the two projections, then define the following effective divisors of { \atop S × P^{2}}:

H̃ ∈ |(π_{1} ∘ α)^{∗}(O_{S}(−K_{S}))| , H̃_{1} ∈ |(π_{1} ∘ α)^{∗}(O_{S}(L))| , H̃_{2} ∈ |(π_{2} ∘ α)^{∗}(O_{P^{2}}(1))|, as well as

Ñ_{i} := α^{−1}(D_{i}) and Ñ = ∑_{i=1}^{4} Ñ_{i}.

Applying push-forward under ϵ_{2}, we obtain the following divisors on P:

H := ϵ_{2∗}(H̃), H_{i} := ϵ_{2∗}(H̃_{i}), N_{i} := ϵ_{2∗}(Ñ_{i}), and N := ∑_{i=1}^{4} N_{i}.

Proposition 2.4. |O_{P}(1)| = |H_{1} + H_{2} − N|. Proof. Using for instance [Ma] Theorem 1.4, we have ϵ_{2}^{∗}(O_{P}(1)) = O{ \atop S×P^{2}}(H̃_{1} + H̃_{2} − Ñ). By pushing forward, we obtain the desired result. □

We have already remarked that h : P → P^{4} is a morphism of degree 2. The inverse image E ⊂ P under h of a general line in P^{4} is a smooth elliptic curve. The restriction h_{E} has 4 branch points and the branch locus of h is a quartic hypersurface B ⊂ P^{4}. Proposition 2.5. For each d ≥ 0, one has h^{0}(P, O_{P}(d)) = \binom{d+4}{4} + \binom{2d}{4}. Proof. We pass to the Stein factorization h := s ∘ f, where f : \overline{P} → P^{4} is a double cover and s : P → \overline{P} is birational. In particular, h^{0}(P, O_{P}(d)) = h^{0}(f^{∗}O_{P}_{4}(d)). The involution ι : \overline{P} → \overline{P}induced by f acts on H^{0}(f^{∗}O_{P}_{4}(d)) and the eigenspaces are f^{∗}H^{0}(O_{P}_{4}(d)) and b · f^{∗}H^{0}(O_{P^{4}}(2d − 4)) respectively, where b ∈ H^{0}(f^{∗}O_{P}_{4}(2)) and div(b) = f^{−1}(B). □

We can now relate the 15-dimensional linear system |O_{P}(2)| of smooth conic bundles in P to the linear system of 4-nodal conic bundles of type (2, 2) in P^{2} × P^{2}. Let Ĩ be the moving part of the total transform ((σ × id_{P^{2}}) ∘ α)^{∗}|I_{{w_{1},...,w_{4}}}^{2}(2, 2)|. Over P, we consider the linear system I′ := (ϵ_{2})_{∗}Ĩ, and conclude that: Proposition 2.6. One has the equality I′ = |O_{P}(2)| of linear systems on P.

Proof. Consider a general threefold Y ∈ |I_{{w_{1},...,w_{4}}}(1, 1)|. Its strict transform Ỹ under the morphism (σ × id_{P^{2}}) ∘ α is smooth and has class H̃_{1} + H̃_{2} − Ñ. Therefore we obtain (ϵ_{2})_{∗}(Ỹ) ∈ |H_{1} + H_{2} − N| = |O_{P}(1)|, and then I′ = |O_{P}(2)|. □

To conclude this discussion, the identification

|O_{P}(2)| = |I_{{w_{1},...w_{2}}}^{2}(2, 2)| := P^{15},

induced by the birational map ϵ, will be used throughout the rest of the paper. Remark 2.7. One can describe h : P → P^{4} in geometric terms. Consider the rational map h′ := h_{D} ∘ (σ × id_{P}_{2})^{−1} : P^{2} × P^{2} −−→ P^{4}, where h_{D} appears in a previous diagram. Then h′ is defined by the linear system |I_{{w_{1},...,w_{4}}}(1, 1)|. If P^{2} × P^{2} ⊂ P^{8} is the Segre embedding and Λ ⊂ P^{8} the linear span of w_{1}, …, w_{4}, then h′ is the restriction to P^{2}×P^{2} of the linear projection having center Λ.

One can also recover the tautological bundle M as follows. Consider the family of planes {P_{x} := h′_{∗}({x} × P^{2})}_{x∈P^{2}}. Its closure in the Grassmannian G(2, 4) of planes of P^{4} is equal to the image of S under the classifying map of M. We omit further details.

Proposition 2.8. The following relations hold in CH^{4}({ \atop S × P^{2}}):

Ñ^{4} = −4, Ñ^{3} · H̃ = 4, Ñ^{3} · H̃_{1} = Ñ^{3} · H̃_{2} = 0, Ñ^{2} · H̃^{2} = Ñ^{2} · H̃_{1}^{2} = Ñ^{2} · H̃_{2}^{2} = 0. Proof. These are standard calculations on blow-ups. We fix i ∈ {1, …, 4} and note that Ñ_{i} = P(O_{D_{i}}^{⊕2} ⊕ O_{D_{i}}(1)). We denote by ξ_{i} := c_{1}(OÑ_{i}(1)) ∈ CH^{1}(Ñ_{i}) the class of the tautological bundle on the exceptional divisor, by α_{i} := α_{|Ñ_{i}} : Ñ_{i} → D_{i} the restriction

of α, and by j_{i} : Ñ_{i} ↪ { \atop S × P^{2}} the inclusion map. Then for k = 1, …, 4, the formula Ñ_{i}^{k} = (−1)^{k−1}(j_{i})_{∗}(ξ_{i}^{k−1}) holds in CH^{k}({ \atop S × P^{2}}). In particular,

Ñ_{i}^{4} = −(j_{i})_{∗}(ξ_{i}^{3}) = −c_{1}(O_{D_{i}}^{⊕2} ⊕ O_{D_{i}}(1)) = −1,

which implies that Ñ^{4} = Ñ_{1}^{4}+⋯+Ñ_{4}^{4} = −4. Furthermore, based on dimension reasons, Ñ_{i}^{2} · α^{∗}(γ) = −(j_{i})_{∗}(ξ_{i} · α_{i}^{∗}(γ_{|D_{i}})) = 0, for each class γ ∈ CH^{2}(S × P^{2}). Finally, for a class γ ∈ CH^{1}(S ×P^{2}), we have that Ñ_{i}^{3} ·α^{∗}(γ) = (j_{i})_{∗}(ξ_{i}^{2} ·α_{i}^{∗}(γ_{|D_{i}})) = (α_{i})_{∗}(ξ_{i}^{2})·γ_{|D_{i}} = γ ·D_{i}, where the last intersection product is computed on S × P^{2}. This determines all top intersection numbers involving Ñ^{3}, which finishes the proof.

□

Remark 2.9. Since ϵ_{2} contracts the divisors { \atop E_{i} × P^{2}}, clearly H = 3H_{1}−N. An immediate consequence of Proposition 2.8 is that the degree of the morphism h : P → P^{4} equals deg(h) = (H_{1} + H_{2} − N)^{4} = 6H_{1}^{2} · H_{2}^{2} + N^{4} = 2.

2.1. Pencils of conic bundles in the projective bundle P. In this section we determine the numerical characters of a pencil of 4-nodal conic bundle of type (2, 2). Let

P ⊂ |O_{P}(2)| = |O_{P}(2H_{1} + 2H_{2} − 2N)|

be a Lefschetz pencil in P. We may assume that its base locus B ⊂ Pis a smooth surface. We are primarily interested in the number of singular conic bundles and those having a double line respectively. We first describe B.

Lemma 2.10. For the base surface B ⊂ P of a pencil of conic bundles, the following hold: (i) K_{B} = O_{B}(H_{1} + H_{2} − N) ∈ Pic(B).

(ii) K_{B}^{2} = 8 and c_{2}(B) = 64.

Proof. The surface B is a complete intersection in P, hence by adjunction

K_{B} = K_{P|B} ⊗ O_{B}(4H_{1} + 4H_{2} − 4N).

Furthermore, K{ \atop S×P^{2}} = α^{∗}(O_{S}(−H) ⊠ O_{P}_{2}(−3)) ⊗ O{ \atop S×P^{2}}(2Ñ), and by push-pull

K_{P} = (ϵ_{2})_{∗}(K{ \atop S×P^{2}}) = O_{P}(−H − 3H_{2} + 2N) = O_{P}(−3H_{1} − 3H_{2} + 3N),

for H = 3H_{1} − N. We find that K_{B} = O_{B}(H_{1} + H_{2} − N). From Lemma 2.8, we compute

K_{B}^{2} = 4(H_{1} + H_{2} − N)^{2} · (H_{1} + H_{2} − N)^{2} = 24H_{1}^{2} · H_{2}^{2} + 4N^{4} = 8.

Finally, from the Euler formula applied for B, we obtain 12χ(B, O_{B}) = K_{B}^{2} + c_{2}(B). Since χ(B, O_{B}) = 6, this yields c_{2}(B) = 64. □

For a variety Z we denote as usual by e(Z) its topological Euler characteristic.

Lemma 2.11. For a general conic bundle Q ∈ |O_{P}(2)|, we have that e(Q) = 4, whereas for conic bundle Q_{0} with a single ordinary quadratic singularity, e(Q_{0}) = 5.

Proof. We fix a conic bundle π_{1} : Q → S with smooth discriminant curve C ∈ | − 2K_{S}|. We then write the relation e(Q − π_{1}^{∗}(C)) = 2e(S − C). Since e(π_{1}^{∗}(C)) = 3e(C), we find that e(Q) = 2e(S) + e(C) = 2 · 7 − 10 = 4.

Similarly, if π_{1} : Q_{0} → P^{2} is a conic bundle such that the discriminant curve C_{0} ⊂ S has a unique node, then e(Q_{0}) = 2e(S) + e(C_{0}) = 14 − 9 = 5. □

In the next statement we use the notation from [FL] for divisors classes on \overline{R}_{g}, see also the beginning of Section 3 for further details.

Theorem 2.12. In a Lefschetz pencil of conic bundles P ⊂ |O_{P}(2)| there are precisely 77 singular conic bundles and 32 conic bundles with a double line.

Proof. Retaining the notation from above, B ⊂ P is the base surface of the pencil. The number δ of nodal conic bundles in P is given by the formula:

δ = e(P) + e(B) − 2e(Q) = 3e(S) + 64 − 2 · 4 = 77,

where the relation e(P) = 3e(S) follows because π : P → S is a P^{2}-bundle.

The number of conic bundles in the pencil P having a double line equals the number of discriminant curves in the family induced by P in \overline{R}_{6}, that lie in the ramification divisor Δ_{0}^{ram} of the projection map π : \overline{R}_{6} → \overline{M}_{6}. We choose general conic bundles Q_{1}, Q_{2} ∈ P, and let A = (a_{ij}(x_{1}, x_{2}, x_{3}))_{i,j=1}^{3} and B = (b_{ij}(x_{1}, x_{2}, x_{3}))_{i,j=1}^{3} be the symmetric matrices of quadratic forms giving rise to Prym curves (Γ_{1}, η_{1}) := 𝔡(Q_{1}) and (Γ_{2}, η_{2}) := 𝔡(Q_{2}) ∈ P_{6} respectively. Note that both curves Γ_{1} and Γ_{2} are nodal precisely at the points u_{1}, …, u_{4}. Let us consider the surface

Y := {([x_{1} : x_{2} : x_{3}], [t_{1} : t_{2}]) ∈ P^{2} × P^{1} : det((t_{1}a_{ij} + t_{2}b_{ij})(x_{1}, x_{2}, x_{3})) = 0},

together with the projection γ : Y → P^{1}. If h_{1}, h_{2} ∈ CH^{1}(P^{2} × P^{1}) are the pull-backs of the hyperplane classes under the two projections, then Y ≡ 6h_{1} + 3h_{2}. Therefore ω_{Y} = O_{Y} (3h_{1} + h_{2}) and h^{0}(Y, O_{Y}) = 20. Observe that the surface Y is singular along the curves L_{j} := {u_{j}} × P^{1} for j = 1, …, 4, and let ν_{Y} : Y → Y be the normalization. From the exact sequence

0 ⟶ H^{0}(Y, ω_{Y}) ⟶ H^{0}(Y, ω_{Y}) ⟶ ⨁_{j=1}^{4} H^{0}(L_{j}, ω_{Y |L_{j}}) ⟶ 0,

taking also into account that ω_{Y |L_{j}} = O_{L_{j}}(1), we compute that h^{0}(Y, O_{Y}) = 12, and hence χ(Y, O_{Y}) = 13. The morphism γ̃ := γ ∘ ν_{Y} : Y → P^{1} is a family of Prym curves of genus 6 and it induces a moduli map m(γ̃) : P^{1} → \overline{R}_{6}.

The points u_{1}, …, u_{4} ∈ P^{2} being general, the curve 𝔢 := m(γ̃)(P^{1}) ⊂ \overline{R}_{6} is disjoint from the pull-back π^{∗}(\overline{GP}_{6}) ⊂ \overline{R}_{6} of the Gieseker-Petri divisor consisting of curves of genus 6 lying on a singular quintic del Pezzo surface, see [FGSMV] for details on the geometry of π^{−1}(\overline{GP}_{6}). Since π^{∗}([\overline{GP}_{6}])_{|R̃_{6}} = 94λ − 12(δ0^{′} + δ0^{′′} + 2δ_{0}^{ram}) ∈ CH^{1}(R̃_{6}), and 𝔢 · δ_{0}^{ram} = 77 (this being the already computed number of nodal conic bundles in P), whereas 𝔢 · δ0^{′} = 0, we obtain the following relation

47𝔢 · λ − 6𝔢 · δ0^{′} − 12𝔢 · δ_{0}^{ram} = 0.

Finally, we observe that 𝔢 · λ = χ(Y, O_{Y}) + g − 1 = 18, which leads to 𝔢 · δ_{0}^{ram} = 32. □

3. A sweeping rational curve in the boundary of \overline{A}_{6}

In this section we construct an explicit sweeping rational curve in C̃^{5}, whose numerical properties we shall use in order to bound the slope of \overline{A}_{6}. Before doing that, we quickly review basic facts concerning the moduli space \overline{R}_{g} of stable Prym curves of genus g, while referring to [FL] for details.

Geometric points of \overline{R}_{g} correspond to triples (X, η, β), where X is a quasi-stable curve of arithmetic genus g, η is a line bundle on X of degree 0, such that η_{E} = O_{E}(1) for each smooth rational component E ⊂ X with |E ∩ \overline{(X − E)}| = 2, and β : η^{⊗2} → O_{X} is a sheaf homomorphism whose restriction to any non-exceptional component of X is an isomorphism. Denoting by π : \overline{R}_{g} → \overline{M}_{g} the forgetful map, one has the following formula [FL] Example 1.4

(5)  π^{∗}(δ_{0}) = δ0^{′} + δ0^{′′} + 2δ_{0}^{ram} ∈ CH^{1}(\overline{R}_{g}),

where δ0^{′} := [Δ0^{′}], δ0^{′′} := [Δ0^{′′}], and δ_{0}^{ram} := [Δ_{0}^{ram}] are boundary divisor classes on \overline{R}_{g} whose meaning we recall. Let us fix a general point [C] ∈ Δ_{0} corresponding to a smooth 2-pointed curve (N, x, y) of genus g − 1 with normalization map ν : N → C, where ν(x) = ν(y). A general point of Δ0^{′} (respectively of Δ0^{′′}) corresponds to a stable Prym curve [C, η], where η ∈ Pic^{0}(C)[2] and ν^{∗}(η) ∈ Pic^{0}(N) is non-trivial (respectively, ν^{∗}(η) = O_{N}). A general point of Δ_{0}^{ram} is of the form (X, η), where X := N ∪_{{x,y}} P^{1} is a quasi-stable curve of arithmetic genus g, whereas η ∈ Pic^{0}(X) is a line bundle characterized by η_{P}_{1} = O_{P}_{1}(1) and η_{N}^{⊗2} = O_{N}(−x − y). Throughout this paper, we only work on the partial compactification R̃_{g} := π^{−1}(M_{g} ∪ Δ_{0}) of R_{g} and ignore the boundary divisors of \overline{R}_{g} corresponding to curves of compact type. We denote by δ0^{′}, δ0^{′′} and δ_{0}^{ram} the restrictions of the corresponding boundary classes to R̃_{g}. Note that

CH^{1}(R̃_{g}) = Q⟨λ, δ0^{′}, δ0^{′′}, δ_{0}^{ram}⟩.

Recall that we use the identification P^{15} := |I_{{w_{1},...,w_{4}}}^{2}(2, 2)| = |O_{P}(2)| for the linear system of (2, 2) threefolds in P^{2} × P^{2} which are nodal at w_{1}, …, w_{4}. Recall also that P → S is the P^{2}-bundle constructed in section 2.

We start constructing a sweeping curve i : P^{1} → C̃^{5}, by fixing general elements (o_{1}, ℓ_{1}), …, (o_{4}, ℓ_{4}) ∈ P^{2} × (P^{2})^{∨} and a general point o ∈ P^{2}. We introduce the net

T := {Q ∈ P^{15} : (o, o) ∈ Q and {o_{j}} × ℓ_{i} ⊂ Q for j = 1, …, 4},

consisting of conic bundles containing the lines {o_{1}} × ℓ_{1}, …, {o_{4}} × ℓ_{4} and passing through the point (o, o) ∈ P^{2}×P^{2}. Because of the genericity of our choices, the restriction

res_{|{o}×P^{2}} : T → |O_{{o}×P^{2}}(2)|

is an injective map and we can view T as a general net of conics in P^{2} passing through the fixed point o ∈ P^{2}. The discriminant curve of the net is a nodal cubic curve Δ_{T} ⊂ T; its singularity corresponds to the only conic of type ℓ_{0} + m_{0} ∈ T, consisting of a pair of lines ℓ_{0} and m_{0} passing through o.

To ease notation, we identify {o}×P^{2} with P^{2} in everything that follows. Denoting by P^{1} := P(T_{o}(P^{2})) the pencil of lines through o, it is clear that the map

τ : P^{1} → Δ_{T} , τ(ℓ) := Q_{ℓ} ∈ T, such that Q_{ℓ} ⊃ {o} × ℓ,

is the normalization map of Δ_{T}. In particular, we have τ(ℓ_{0}) = τ(m_{0}) = ℓ_{0} + m_{0}, where, abusing notation, we identify Q_{ℓ} with its singular conic {o} × (ℓ + m) = Q_{ℓ} · ({o} × P^{2}). For ℓ ∈ P^{1}, the double cover f_{ℓ} : Γ̃_{ℓ} → Γ_{ℓ} over the discriminant curve Γ_{ℓ} of Q_{ℓ} is an element of P_{6} (see Definition 1.1). Clearly Γ̃_{ℓ} carries the marked points ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4} and ℓ. This procedure induces a moduli map into the universal symmetric product

i : P^{1} → C̃^{5}, i(ℓ) := [ρ(Γ̃_{ℓ}/Γ_{ℓ}), ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ].

We explicitly construct the family of discriminant curves Γ_{ℓ} of the conic bundles Q_{ℓ}, where τ(ℓ) ∈ Δ_{T}. Setting coordinates x := [x_{1} : x_{2} : x_{3}], y := [y_{1} : y_{2} : y_{3}] in P^{2}, let

Z := {(x, y, t) ∈ P^{2} × P^{2} × Δ_{T} : y ∈ Sing(π_{1}^{−1}(x) ∩ Q_{t})} ⊂ P^{2} × P^{2} × T.

Concretely, if Q_{1}, Q_{2}, Q_{3} is a basis of T, then the surface Z is given by the equations

\frac{∂}{∂y_{i}} (t_{1}Q_{1}(x, y) + t_{2}Q_{2}(x, y) + t_{3}Q_{3}(x, y)) = 0, for i = 1, 2, 3,

where [t_{1} : t_{2} : t_{3}] ∈ T. It follows immediately that Z is a complete intersection of three divisors of multidegree (2, 1, 1), defined by the partial derivatives, and the divisor P^{2} × P^{2} × Δ_{T} of multidegree (0, 0, 3).

Lemma 3.1. The first projection γ_{1} : Z → P^{2} is a map of degree 9.

Proof. Denoting by h_{1}, h_{2}, h_{3} ∈ Pic(P^{2}×P^{2}×T) the pull-backs of the hyperplane bundles from the three factors, we find that deg(γ_{1}) = (2h_{1} + h_{2} + h_{3})^{3} · (3h_{3}) · h_{1}^{2} = 9. □

The third projection γ_{3} : Z → Δ_{T} is a birational model of the family {C_{ℓ}}_{ℓ∈P^{1}} of underlying genus 6 curves, induced by the map i : P^{1} → C̃^{5}. However, the surface Z is not normal. It has singularities along the curves {w_{j}} × Δ_{T} for j = 1, …, 4, as well as along the fibre γ_{3}^{−1}(ℓ_{0} + m_{0}) over the point τ(ℓ_{0}) = τ(m_{0}) = ℓ_{0} + m_{0} ∈ T. To construct a smooth model of Z, we pass instead to its natural birational model in the 5-fold P× P^{1}.

Abusing notation, we still denote by Q_{ℓ} ⊂ P the strict transform of the conic bundle Q_{ℓ} in P^{2} × P^{2}; its discriminant curve C_{ℓ} is viewed an an element of | − K_{S}|. We denote by π_{ℓ} : Q_{ℓ} → S the restriction of π : P → S, then consider the surface

Z := {(z, ℓ) ∈ P× P^{1} : z ∈ Sing π_{ℓ}^{−1}(C_{ℓ})}.

Clearly Z is endowed with the projection q_{P^{1}} : Z → P^{1}. We have the following commutative diagram, where u := (σ ×id_{P^{2}})∘ϵ^{−1} : P → P^{2} ×P^{2} and the horizontal arrows are the natural inclusions or projections and ν_{Z} : Z → Z is the normalization map:

\xymatrix{Z \ar[r] \ar[rr]^{q_{_{P}1}} \ar[d]_{ν_{Z}} & P× P^{1} \ar[d]^{u×τ} \ar[r] & P^{1} \ar[d]_{τ} \\ Z \ar[r] \ar[rr]_{γ_{3}} & P^{2} × P^{2} × Δ_{T} \ar[r] & Δ_{T}}

Since u × τ is birational, it follows that deg(Z/S) = deg(Z/P^{2}) = 9. The fibration q_{P^{1}} : Z → P^{1} admits sections

τ_{j} : P^{1} → Z for j = 1, …, 5,

which we now define. For 1 ≤ j ≤ 4 and each ℓ ∈ P^{1}, the fibre Q_{ℓ} · ({o_{j}} × P^{2}) contains the line ℓ_{j}. Hence ℓ_{j} defines a point in the covering curve of f_{ℓ} : C̃_{ℓ} → C_{ℓ}. By definition τ_{j}(ℓ) is this point. Tautologically, τ_{5}(ℓ) is the point corresponding to the line ℓ.

Finally, we consider the universal family Q ⊂ P× T defined by T. The pull-back of the projection Q → T by the morphism id_{P}× τ induces a flat family of conic bundles Q′ ⊂ P× P^{1} and a projection q′ : Q′ → P^{1}. Clearly, Z ⊂ Q′ and q_{P}_{1} = q′_{|Z}.

Definition 3.2. A conic bundle Q ∈ |O_{P}(2)| is said to be ordinary if both Q and its discriminant cover curve C are nodal. A subvariety in |O_{P}(2)| is said to be a Lefschetz family, if each of its members is an ordinary conic bundles.

Postponing the proof, we assume that the fibration q′ : Q → P^{1} constructed above is a Lefschetz family of conic bundles, and we determine the properties of the Prym moduli map m : P^{1} → \overline{R}_{6}, where m(ℓ) := [f_{ℓ} : C̃_{ℓ} → C_{ℓ}] = ρ(Γ̃_{ℓ}/Γ_{ℓ}).

Proposition 3.3. The numerical features of m : P^{1} → \overline{R}_{6} are as follows:

m(P^{1}) · λ = 9 · 6, m(P^{1}) · δ0^{′} = 3 · 77, m(P^{1}) · δ_{0}^{ram} = 3 · 32, m(P^{1}) · δ0^{′′} = 0.

Proof. We consider the composite map ρ ∘ 𝔡_{|T} : T −−→ \overline{R}_{6}, assigning to a conic bundle from the net T ⊂ P^{15} the double covering of its (normalized) discriminant curve. This map is well-defined outside the codimension two locus in T corresponding to conic bundles with non-nodal discriminant. Furthermore, m = ρ ∘ 𝔡 ∘ τ : P^{1} → \overline{R}_{6}, where we recall that τ(P^{1}) = Δ_{T} ⊂ T is a nodal cubic curve. It follows that the intersection number of m(P^{1}) ⊂ \overline{R}_{6} with any divisor class on \overline{R}_{6} is three times the intersection number of the corresponding class in CH^{1}(\overline{R}_{6}) with the curve of discriminants induced by a pencil of conic bundles in |O_{P}(2)|. The latter numbers have been determined in Theorem 2.12. □

The composition of the map i : P^{1} → C̃^{5} with the projection C̃^{5} → \overline{R}_{6} is the map m : P^{1} → \overline{R}_{6} discussed in Proposition 3.3. We discuss the numerical properties of i:

Proposition 3.4. The moduli map i : P^{1} → C̃^{5} induced by the pointed family of Prym curves

(q_{P}_{1} : Z → P^{1}, τ_{1}, …, τ_{5} : P^{1} → Z)

sweeps the five-fold product C̃^{5}. Furthermore i(P^{1}) · ψ_{x_{j}} = 9, for j = 1, …, 5.

Proof. For 1 ≤ j ≤ 4, the image of the section τ̃_{j} := ν_{Z} ∘ τ_{j} : P^{1} → Z is the curve

L_{j} := {(o_{j}, y_{j}(ℓ), ν(ℓ)) ∈ P^{2} × P^{2} × T : ℓ ∈ P^{1}},

where y_{j}(ℓ) = ℓ_{j} ∩ m_{j}(ℓ), with m_{j}(ℓ) being the line in P^{2} defined by the equality of cycles Q_{ℓ} · ({o_{j}} × P^{2}) = {o_{j}} × (ℓ_{j} + m_{j}(ℓ)). Here, recall that ℓ ∈ P^{1} = P(T_{o}(P^{2}) is a point corresponding to a line in P^{2} passing through o. In particular, noting that by the adjunction formula ω_{Z} = O_{Z}(3h_{1} + 3h_{3}), we compute that L_{j} · h_{1} = 0 and L_{j} · h_{3} = 3, hence L_{j} · ω_{Z} = L_{j} · (3h_{1} + 3h_{3}) = 9.

By definition i(P^{1}) · ψ_{x_{j}} = τ_{j}^{∗}(c_{1}(ω_{q_{_{P}1}} )). To evaluate the dualizing class, we note that ω_{q_{_{P}1}} = ω_{Z} ⊗ q_{P^{1}}^{∗}(T_{P}_{1}), therefore deg τ_{j}^{∗}(c_{1}(ω_{q_{_{P}1}} )) = deg τ_{j}^{∗}(ω_{Z}) + 2. Furthermore,

ν_{Z}^{∗}(ω_{Z}) = ω_{Z} ⊗ O_{Z}(q_{P^{1}}^{−1}(ℓ_{0}) + q_{P^{1}}^{−1}(m_{0}) + D),

where D ⊂ Z is a curve disjoint from ν_{Z}^{−1}(L_{j}). We compute that

deg τ_{j}^{∗}(ω_{Z}) = deg τ̃_{j}^{∗}(ω_{Z}) − deg τ_{j}^{∗}q_{P^{1}}^{∗}(ℓ_{0}) − deg τ_{j}^{∗}q_{P^{1}}^{∗}(m_{0}) = ω_{Z} · L_{j} − 2,

and finally, i(P_{1}) · ψ_{x_{j}} = ω_{Z} · L_{j} = 9. The calculation of i(P^{1}) · ψ_{x_{5}} is largely similar and we skip it. □

Proof of the claim.We show that q′ : Q → P^{1} is a Lefschetz family, that is, it consists entirely of ordinary conic bundles. For 1 ≤ j ≤ 4, let ℓ′_{j} ⊂ Pbe the inverse image of the line {o_{j}} × ℓ_{j} under the map u : P −−→ P^{2} × P^{2} and set W := |I_{{ℓ′_{1},...,ℓ′_{4}}}(2)| ⊂ |O_{P}(2)|. The net T := T_{o} of conic bundles passing through the point (o, o) ∈ P^{2} × P^{2} is a plane in W. Let Δ_{no} denote the locus of non-ordinary conic bundles Q ∈ W. We aim to show that Δ_{no} ∩ Δ_{T_{o}} = ∅, for a general point o ∈ P^{2}.

We consider the incidence correspondence

Σ := {(Q, (o, ℓ)) ∈ Δ_{no} × G : {o} × ℓ ⊂ u(Q), o ∈ ℓ}

together with the projection map p_{1} : Σ → Δ_{no}. Over a conic bundle Q ∈ Δ_{no} for which the image u(Q) ⊂ P^{2} × P^{2} is transversal to a general fibre {o} × P^{2}, the fibre p_{1}^{−1}(Q) is finite. To account for the conic bundles not enjoying this property, we define Δ_{hr} to be the union of the irreducible components of Δ_{no} consisting of conic bundles Q ∈ W such that the branch locus of Q → S is equal to S.

To conclude that Δ_{no} ∩ Δ_{T_{o}} = ∅ for a general o ∈ P^{2}, it suffices to show that (1) Δ_{no} has codimension at least 2 in W, and (2) Δ_{hr} has codimension at least 3 in W. The next two lemmas are devoted to the proof of these assertions. □

Lemma 3.5. Δ_{no} has codimension at least 2 in W.

Proof. We have established that h : P → P^{4} is a morphism of degree two. We claim that the 4 lines l_{i} := h(ℓ′_{i}) ⊂ P^{4} are general, in the sense that V := |I_{{l_{1},...,l_{4}}}(2)| is a net of quadrics. Granting this and denoting by L_{ij} ∈ H^{0}(P^{4}, O_{P}_{4}(1)) the linear form vanishing along l_{i} ∪ l_{j}, the space V is generated by the quadrics L_{12} · L_{34}, L_{13} · L_{24} and L_{14} · L_{23} respectively. The base locus bs |V | of the net is a degenerate canonical curve of genus 5, which is a union of 8 lines, namely l_{1}, …, l_{4} and b_{1}, …, b_{4}, where if {1, 2, 3, 4} = {i, j, k, l}, then the line b_{l} ⊂ P^{4} is the common transversal to the lines l_{i}, l_{j} and l_{k}. Then by direct calculation, the pull-back P of a general pencil in V is a Lefschetz family of conic budles in |O_{P}(2)|. Since P ∩ Δ_{no} = ∅, it follows that codim(Δ_{no}, W) ≥ 2. It remains to show that the lines l_{1}, …, l_{4} are general. To that end, we observe that the construction can be reversed. Four general lines m_{1}, …, m_{4} ∈ G(1, 4) ⊂ P^{9} define a codimension 4 linear section S′ of G(1, 4) which is isomorphic to S. The projectivized universal bundle P′ → S′ is a copy of Pand the projection h′ : P′ → P^{4} is the tautological map. This completes the proof. □

The second lemma follows from a direct analysis in P^{2} × P^{2}.

Lemma 3.6. Δ_{hr} has codimension at least 3 in W.

Proof. If Q is a general element of an irreducible component of Δ_{hr} , then the discriminant locus of the projection p : Q → S equals S, and necessarily Q = D + D′, where p(D) = p(D′) = S. By a dimension count, it follows that W contains only finitely many elements Q ∈ Δ_{hr}, such that D, D′ ∈ |O_{P}(1)|, and assume that we are not in this case.

Recall that h′ : P^{2} × P^{2} −−→ P^{4} is the map defined by |I_{{w_{1},...,w_{4}}}(1, 1)|. The case when both u(D), u(D′) ∈ |I_{{w_{1},...,w_{4}}}(1, 1)| having been excluded, we may assume that one of the components of u(Q), say u(D) ⊂ P^{2} × P^{2}, has type (0, 1). In particular, u(D) = P^{2} × n, where n ⊂ P^{2} is a line. Observe that u(D) has degree three in the Segre embedding P^{2}×P^{2} ⊂ P^{8} and the base scheme of |I_{{w_{1},...,w_{4}}}(1, 1)| consists of the simple points w_{1}, …, w_{4}. Since h′(D) lies on a quadric, it follows u(D) ∩ {w_{1}, …, w_{4}} ≠ ∅, therefore we have u_{i} ∈ n for some i, say i = 4. Since the lines {o_{i}} × ℓ_{i} are general, they do not lie on u(D), for ℓ_{i} ≠ n. Hence u(D′) ⊂ P^{2} × P^{2} is a (2, 1) hypersurface which contains {o_{1}} × ℓ_{1}, … {o_{4}} × ℓ_{4}, is singular at w_{1}, w_{2}, w_{3} and such that w_{4} ∈ u(D′). This contradicts the generality of the lines {o_{i}} × ℓ_{i}. □

4. The slope of \overline{A}_{6}

For g ≥ 2, let \overline{A}_{g} be the first Voronoi compactification of A_{g} — this is the toroidal compactification of A_{g} constructed using the perfect fan decomposition, see [SB]. The rational Picard group of \overline{A}_{g} has rank 2 and it is generated by the first Chern class λ_{1} of the Hodge bundle and the class of the irreducible boundary divisor D = D_{g} := \overline{A}_{g} − A_{g}. Following Mumford [Mu], we consider the moduli space Ã_{g} of principally polarized abelian varieties of dimension g together with their rank 1 degenerations. Precisely, if ξ : \overline{A}_{g} → A_{g}^{s} = A_{g} ⊔ A_{g−1} ⊔ … ⊔ A_{1} ⊔ A_{0} is the projection from the toroidal to the Satake compactification of A_{g}, then

Ã_{g} := \overline{A}_{g} − ξ^{−1}(⋃_{j=2}^{g} A_{g−j}) := A_{g} ⊔ D̃_{g},

where D̃_{g} is an open dense subvariety of D_{g} isomorphic to the universal Kummer variety Kum(X_{g−1}) := X_{g−1}/±. Furthermore, if ϕ : X̃_{g−1} → Ã_{g−1} is the extended universal abelian variety, there exists a degree two morphism j : X̃_{g−1} → \overline{A}_{g}, extending the Kummer map X̃_{g−1} →^{2:1} D̃_{g}. The geometry of the boundary divisor ∂X̃_{g−1} = ϕ^{−1}(Kum(X_{g−2})) is discussed in [vdG] and [EGH]. In particular, codim(D_{g} − j(∂X̃_{g−1}), D_{g}) = 2. As usual, let 𝔼_{g} be the Hodge bundle on \overline{A}_{g}.

Denoting by φ : Ỹ_{g} → R̃_{g} the universal Prym variety restricted to the partial compactification R̃_{g} of \overline{R}_{g} introduced in Section 3, we have the following commutative diagram summarizing the situation, where the lower horizontal arrow is the Prym map:

\xymatrix{Ỹ_{g} \ar[r]^{χ} \ar[d]_{φ} & X̃_{g−1} \ar[d]_{ϕ} \ar[r]^{j} & \overline{A}_{g} \\ R̃_{g} \ar[r]^{P} & Ã_{g−1}}

Furthermore, let us denote by θ ∈ CH^{1}(X̃_{g−1}) the class of the universal theta divisor trivialized along the zero section and by θ_{pr} := χ^{∗}(θ) ∈ CH^{1}(Ỹ_{g}) the Prym theta divisor. The following formulas have been pointed out to us by Sam Grushevsky:

Proposition 4.1. The following relations at the level of divisor classes hold:

(i) j^{∗}([D]) = −2θ + ϕ^{∗}([D_{g−1}]) ∈ CH^{1}(X̃_{g−1}). (ii) (j ∘ χ)^{∗}(λ_{1}) = φ^{∗}(λ − \frac{1}{4}δ_{0}^{ram}) ∈ CH^{1}(Ỹ_{g}).

(iii) (j ∘ χ)^{∗}([D]) = −2θ_{pr} + φ^{∗}(δ0^{′}) ∈ CH^{1}(Ỹ_{g}).

Proof. At the level of the restriction j : X_{g−1} → \overline{A}_{g}, the formula

j^{∗}(D) ≡ −2θ ∈ CH^{1}(X_{g−1})

is proven in [Mu] Proposition 1.8. To extend this calculation to X̃_{g−1}, it suffices to observe that the boundary divisor ∂X̃_{g−1} = ϕ^{∗}(D̃_{g−1}) is mapped under j to the locus in \overline{A}_{g} parametrizing rank 2 degenerations and it will appear with multiplicity one in j^{∗}(D).

To establish relation (ii), we observe that j^{∗}(λ_{1}) = ϕ^{∗}(λ_{1}), where we use the same symbol to denote the Hodge class on A_{g} and that on A_{g−1}. Indeed, there exists an exact sequence of vector bundles on X̃_{g}, see also [vdG] p.74:

0 ⟶ ϕ^{∗}(𝔼_{g−1}) ⟶ j^{∗}(𝔼_{g}) ⟶ OX̃_{g−1} ⟶ 0.

It follows that χ^{∗}j^{∗}(λ_{1}) = φ^{∗}P^{∗}(λ_{1}) = φ^{∗}(λ − \frac{1}{4}δ_{0}^{ram}), where P^{∗}(λ_{1}) = λ − \frac{1}{4}δ_{0}^{ram}, see [FL], [GSM]. Finally, (iii) is a consequence of (i) and of the relation P^{∗}([D̃_{g−1}]) = δ0^{′}, see [GSM]. □

Assume now that g is an even integer and let π̃ : C̃ → \overline{R}_{g} be the universal curve of genus 2g − 1, that is, C̃ = \overline{M}_{2g−1,1} ×_{\overline{M}_{2g−1}} \overline{R}_{g}, and π : \overline{C} → \overline{R}_{g} the universal curve of

genus g, that is, \overline{C} = \overline{M}_{g,1} ×_{\overline{M}_{g}} \overline{R}_{g}. There is a degree two map f : C̃ → \overline{C} unramified in

codimension one and an involution ι : C̃ → C̃, such that f ∘ι = f. Note that ωπ̃ = f^{∗}(ω_{π}).

We consider the global Abel-Prym map 𝔞𝔭 : C̃^{g−1} −−→ Ỹ_{g}, defined by

𝔞𝔭(C̃/C, x_{1}, …, x_{g−1}) := (C̃/C, OC̃(x_{1}−ι(x_{1})+⋯+x_{g−2}−ι(x_{g−2})+2x_{g−1}−2ι(x_{g−1}))).

Remark 4.2. We recall that if C̃ → C is an étale double cover and ι : C̃ → C̃ the induced involution, then the Prym variety P (C̃/C) ⊂ Pic^{0}(C̃) can be realized as the locus of line bundles OC̃(E − ι(E)), where E is a divisor on C̃ having even degree, see [B3]. Furthermore, for a general point [C̃ → C] ∈ \overline{R}_{g}, where g ≥ 3, and for an integer 1 ≤ n ≤ g − 1, the difference map C̃_{n} → Pic^{0}(C̃) given by E ↦ OC̃(E − ι(E)) is generically finite. In particular, for even g, the locus

Z_{g−2}(C̃/C) := {OC̃(E − ι(E)) : E ∈ C̃_{g−2}}

is a divisor inside P (C̃/C). We refer to Z_{g−2}(C̃/C) as the top difference Prym variety.

One computes the pull-back of the universal theta divisor under the Abel-Prym map. Recall that ψ_{x_{1}}, …, ψ_{x_{g−1}} ∈ CH^{1}(C̃^{g−1}) are the cotangent classes corresponding to the marked points on the curves of genus 2g − 1.

Proposition 4.3. For even g, if μ = φ ∘ 𝔞𝔭 : C̃^{g−1} → R̃_{g} denotes the projection map, one has

𝔞𝔭^{∗}(θ_{pr}) = \frac{1}{2} ∑_{j=1}^{g−2} ψ_{x_{j}} + 2ψ_{x_{g−1}} + 0 · (λ + μ^{∗}(δ0^{′} + δ0^{′′} + δ_{0}^{ram})) − ⋯ ∈ CH^{1}(C̃^{g−1}).

Proof. We factor the map 𝔞𝔭 : C̃^{g−1} −−→ Ỹ_{g} as 𝔞𝔭 = 𝔞𝔧 ∘ Δ, where Δ : C̃^{g−1} → C̃^{2g−2} is defined by (x_{1}, …, x_{g−1}) ↦ (x_{1}, …, x_{g−1}, ι(x_{1}), …, ι(x_{g−1})) and 𝔞𝔧 : C̃^{2g−2} −−→ \overline{𝔓𝔦𝔠}_{2g−1}^{0} is the difference Abel-Jacobi map between the first and the last g − 1 marked points on each curve into the universal Jacobian of degree zero over \overline{M}_{2g−1} respectively . There is a generically injective rational map Ỹ_{g} −−→ \overline{𝔓𝔦𝔠}_{2g−1}^{0}, which globalizes the usual inclusion P (C̃/C) ⊂ Pic^{0}(C̃) valid for each Prym curve [C̃ → C] ∈ R_{g}. Using [GZ] Theorem 6, one computes the pull-back 𝔞𝔧^{∗}(θ_{2g−1}) ∈ CH^{1}(C̃^{2g−2}) of the universal theta divisor θ_{2g−1} on \overline{𝔓𝔦𝔠}_{2g−1}^{0} trivialized along the zero section. Remarkably, the coefficient of λ, as well as that of the δ0^{′}, δ0^{′′} and δ_{0}^{ram} classes in this expression, are all zero. This is then pulled-back to C̃^{g−1} keeping in mind that the pull-back of θ_{2g−1} to Ỹ_{g} is equal to 2θ_{pr}. Using the formulas Δ^{∗}(ψ_{x_{j}}) = Δ^{∗}(ψ_{y_{j}}) = ψ_{x_{j}}, and Δ^{∗}(δ_{0:x_{i}y_{j}}) = δ_{0:x_{i}x_{j}}, as well as Δ^{∗}(δ_{0:y_{i}y_{j}}) = δ_{0:x_{i}x_{j}}, we conclude. □

Remark 4.4. The other boundary coefficients of 𝔞𝔭^{∗}(θ_{pr}) ∈ CH^{1}(C̃^{g−1}) can be determined explicitly, but play no role in our future considerations.

Remark 4.5. Restricting ourselves to even g, we consider the restricted (non-dominant) Abel-Prym map 𝔞𝔭_{g−2} : C̃^{g−2} −−→ Ỹ_{g} given by

𝔞𝔭_{g−2}(C̃/C, x_{1}, …, x_{g−2}) := (C̃/C, OC̃((x_{1} − ι(x_{1}) + ⋯ + x_{g−2} − ι(x_{g−2}))),

and obtain the formula: 𝔞𝔭_{g−2}^{∗}(θ_{pr}) = \frac{1}{2} ∑_{j=1}^{g−2} ψ_{x_{j}} + 0 · (λ + μ^{∗}(δ0^{′} + δ0^{′′} + δ_{0}^{ram})) − · · · .

The image of 𝔞𝔭_{g−2} is a divisor Z_{g−2} on Ỹ_{g} characterized by the property

(Z_{g−2})_{|P (C̃/C)} = Z_{g−2}(C̃/C),

for each [C̃ → C] ∈ R_{g}. In other words, Z_{g−2} is the divisor cutting out on each Prym variety the top difference variety. A similar difference variety inside the universal Jacobian over \overline{M}_{g} has been studied in [FV]. Specializing to the case g = 6, the locus

U_{4} := \overline{(j ∘ χ)(Z_{4})} ⊂ \overline{A}_{6}

is a codimension two cycle on \overline{A}_{6}, which will appear as an obstruction for an effective divisor on \overline{A}_{6} to have small slope.

We use these considerations to bound from below the slope of \overline{A}_{6}. Proof of Theorem 0.4. We have seen that the boundary divisor D_{6} of \overline{A}_{6} is filled-up by rational curves h : P^{1} → D_{6} constructed in Theorem 3.4 by pushing-forward the sweeping rational curve i : P^{1} → C̃^{5} of discriminants of a pencil of conic bundles. In particular, γ := h_{∗}(P^{1}) ∈ NE_{1}(\overline{A}_{6}) is an effective class that intersects every non-boundary effective divisor on \overline{A}_{6} non-negatively. We compute using Propositions 4.1 and 4.3:

γ · λ_{1} = i_{∗}(P^{1}) · μ^{∗}(λ − \frac{1}{4}δ_{0}^{ram}) = 6 · 9 − \frac{3 · 32}{4} = 30, and

γ · [D_{6}] = −i_{∗}(P^{1}) · (∑_{j=1}^{4} ψ_{x_{j}} + 4ψ_{x_{5}}) + i_{∗}(P^{1}) · μ^{∗}(δ0^{′}) = −8 · 9 + 3 · 77 = 159.

We obtain the bound s(\overline{A}_{6}) ≥ \frac{γ·[D_{6}]}{γ·λ_{1}} = \frac{53}{10}. □

For effective divisors on \overline{A}_{6} transversal to U_{4}, we obtain a better slope bound:

Theorem 4.6. If E is an effective divisor on \overline{A}_{6} not containing the universal codimension two Prym difference variety U_{4} ⊂ \overline{A}_{6}, then s(E) ≥ \frac{13}{2}.

Proof. We consider the family (q_{P}_{1} : Z → P^{1}, τ_{1}, …, τ_{4} : P^{1} → Z) obtained from the construction explained in Theorem 3.4, where we retain only the first four sections. We obtain an induced moduli map i_{4} : P^{1} → C̃^{4}. Pushing i_{4} forward via the Abel-Prym map, we obtain a curve h_{4} : P^{1} → U_{4} ⊂ \overline{A}_{6}, which fills-up the locus U_{4}. Thus γ_{4} := (h_{4})_{∗}(P^{1}) ∈ NE_{1}(\overline{A}_{6}) is an effective class which intersects non-negatively any effective divisor on \overline{A}_{6} not containing U_{4}. We compute using Theorems 3.3 and 3.4:

γ_{4} · λ_{1} = γ · λ_{1} = 30 and γ_{4} · [D_{6}] = −4 · 9 + 3 · 77 = 195.
□

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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, Dipartimento di Matematica, Largo San Leonardo Murialdo 1-00146 Roma, Italy

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