THE UNIVERSAL THETA DIVISOR OVER THE MODULI SPACE OF CURVES

GAVRIL FARKAS AND ALESSANDRO VERRA

The universal theta divisor over the moduli space A_{g} of principally polarized abelian varieties of dimension g, is the divisor Θ_{g} inside the universal abelian variety X_{g} over A_{g}, characterized by two properties: (i) Θ_{g}_{|[A,Θ]} = Θ, for every principally polarized abelian variety [A, Θ] ∈ A_{g}, and (ii) the restriction s^{∗}(Θ_{g}) along the zero section s : A_{g} → X_{g} is trivial on A_{g}. The study of the geometry of Θ_{g} closely mirrors that of A_{g} itself. Thus it is known that Θ_{g} is unirational for g ≤ 4; the case g ≤ 3 is classical, for g = 4, we refer to [Ve]. The geometry of Θ_{5} will be addressed in the forthcoming paper [FV3]. Whenever A_{g} is of general type (that is, in the range g ≥ 7, cf. [Fr], [Mum], [T]), one can use Viehweg’s additivity theorem [Vi] for the fibre space Θ_{g} → A_{g} whose generic fibre is a variety of general type, to conclude that Θ_{g} is of general type as well. The Kodaira dimension of Θ_{6} (and that of A_{6}) is unknown.

The main aim of this paper is to present a complete birational classification by Kodaira dimension of the universal theta divisor

𝔗𝔥_{g} := M_{g} ×_{A_{g}} Θ_{g}

over the moduli space of curves. If [C] ∈ M_{g} is a smooth curve, the Abel-Jacobi map C_{g−1} → Pic^{g−1}(C) provides a resolution of singularities of the theta divisor Θ_{C} of the Jacobian of C. Thus one may regard the degree g − 1 universal symmetric product \overline{C}_{g,g−1} := \overline{M}_{g,g−1}/𝔖_{g−1} as a birational model of 𝔗𝔥_{g} (having only finite quotient singularities), and ask for the place of 𝔗𝔥_{g} in the classification of varieties. We provide a complete answer to this question. For small genus, 𝔗𝔥_{g} enjoys rationality properties:

Theorem 0.1. 𝔗𝔥_{g} is unirational for g ≤ 9 and uniruled for g ≤ 11.

The first part of the theorem, is a consequence of Mukai’s work [M1], [M2] on representing canonical curves with general moduli as linear sections of certain homogeneous varieties. When g ≤ 9, there exists a Fano variety V_{g} ⊂ P^{N_{g}} of dimension n_{g} := N_{g} − g + 2 and index n_{g} − 2, such that general 1-dimensional complete intersections of V_{g} are canonical curves [C] ∈ M_{g} having general moduli. The correspondence

Σ := {((x_{1}, …, x_{g−1}), Λ) ∈ V_{g}^{g−1} × G(g, N_{g} + 1) : x_{i} ∈ Λ, for i = 1, …, g − 1}

maps dominantly onto 𝔗𝔥_{g} via the map ((x_{1}, …, x_{g−1}), Λ) ↦ [V_{g} ∩ Λ, x_{1} + ⋯ + x_{g−1}]. Since Σ is a Grassmann bundle over the rational variety V_{g}^{g−1}, it follows that 𝔗𝔥_{g} is unirational in the range g ≤ 9. The cases g = 10, 11 are settled by the observation that in this range the space \overline{M}_{g,g−1} is uniruled, see [FP], [FV2].

For the remaining genera, we prove the following classification result:

Theorem 0.2. The universal theta divisor 𝔗𝔥_{g} is a variety of general type for g ≥ 12.

We also have a birational classification theorem for the universal degree n symmetric product \overline{C}_{g,n} := \overline{M}_{g,n}/𝔖_{n} for all 1 ≤ n ≤ g − 2, and refer to Section 3 for details.

Our results are complete in degree g − 2 and less precise as n decreases. Similarly to Theorem 0.2, the nature of \overline{C}_{g,g−2} changes when g = 12:

Theorem 0.3. The universal degree g − 2 symmetric product \overline{C}_{g,g−2} is uniruled for g < 12 and a variety of general type for g ≥ 12.

The proofs of Theorems 0.2 and 0.3 rely on two ingredients. First, we use our result [FV2], stating that for g ≥ 4, the singularities of \overline{C}_{g,n} impose no adjoint conditions, that is, pluricanonical forms defined on the smooth locus of \overline{C}_{g,n} extend to a smooth model of the symmetric product. Precisely, if ϵ : C̃_{g,n} → \overline{C}_{g,n} denotes any resolution of singularities, then for any l ≥ 0, there is a group isomorphism

ϵ^{∗} : H^{0}((\overline{C}_{g,n})_{reg}, K\overline{C}{⊗l \atop g,n}) →^{≅} H^{0}(C̃_{g,n}, KC̃_{g,n}^{⊗l}).

In particular, 𝔗𝔥_{g} is of general type when the canonical class K_{\overline{C}_{g,g−1}} ∈ Pic(\overline{C}_{g,g−1}) is big. This makes the problem of understanding the effective cone of \overline{C}_{g,g−1} of some importance. If π : \overline{M}_{g,g−1} → \overline{C}_{g,g−1} is the quotient map, the Hurwitz formula gives that

(1)  π^{∗}(K_{\overline{C}_{g,g−1}}) ≡ K_{\overline{M}_{g,g−1}} − δ_{0:2} ∈ Pic(\overline{M}_{g,g−1}).

The sum ∑_{i=1}^{g−1} ψ_{i} ∈ Pic(\overline{M}_{g,g−1})^{𝔖_{g−1}} of cotangent tautological classes descends to a big and nef class on \overline{C}_{g,g−1} (cf. Proposition 1.2), thus in order to conclude that 𝔗𝔥_{g} is of general type, it suffices to exhibit an effective divisor 𝔇 ∈ Eff(\overline{C}_{g,g−1}), such that

(2) π^{∗}(K_{\overline{C}_{g,g−1}}) ∈ ℚ_{>0}⟨λ, ∑_{i=1}^{g−1} ψ_{i}⟩ + ϕ^{∗}Eff(\overline{M}_{g}) + ℚ_{≥0}⟨π^{∗}([𝔇]), δ_{i:c} : i ≥ 0, c ≥ 2⟩.

In this formula, ϕ : \overline{M}_{g,g−1} → \overline{M}_{g} denotes the morphism forgetting the marked points, and refer to Section 1 for the standard notation for boundary divisor classes on \overline{M}_{g,n}. Comparing condition (2) against the formula for K_{\overline{C}_{g,g−1}} given by (4), if one writes π^{∗}(𝔇) ≡ aλ − b_{irr}δ_{irr} + c ∑_{i=1}^{g−1} ψ_{i} − ∑_{i,c} b_{i:c}δ_{i:c} ∈ Pic(\overline{M}_{g,g−1}), the following inequality

(3)  3c < b_{0:2}

is a necessary condition for the existence of a divisor 𝔇 satisfying (2). It is straightforward to unravel the geometric significance of the condition (3). If [C] ∈ M_{g} is a general curve, there is a rational map u : C_{g−1} −−→ \overline{C}_{g,g−1} given by restriction. Denoting by x, θ ∈ N^{1}(C_{g−1})_{ℚ} the standard generators of the Néron-Severi group of the symmetric product, the inequality (3) characterizes precisely those divisors 𝔇 ∈ Pic(\overline{C}_{g,g−1}) for which u^{∗}([𝔇]) lies in the fourth quarter of the (θ, x)-plane (see [K1] for details on the effective cone of C_{g−1}). The divisor 𝔇 ⊂ \overline{C}_{g,g−1} playing this role in our case, is the residual divisor of the universal ramification locus of the Gauss map.

For a curve [C] ∈ M_{g}, we denote by γ : C_{g−1} −−→ (P^{g−1})^{∨} the Gauss map, given by γ(D) := ⟨D⟩ for D ∈ C_{g−1} − C_{g−1}^{1}. The branch divisor Br_{C}(γ) ⊂ (P^{g−1})^{∨} is isomorphic to the dual of the canonical curve C ⊂ P^{g−1}. The closure in C_{g−1} of the ramification divisor Ram_{C}(γ) is isomorphic to the diagonal Δ_{C} := {2p + D : p ∈ C, D ∈ C_{g−3}}, see [An]. In particular, this identification allows one to reconstruct the curve C from the theta divisor Θ_{C} and thus prove Torelli’s theorem. Let us consider the residual divisor Res_{C}(γ), defined via the following equality of divisors on C_{g−1}

γ^{∗}(Br_{C}(γ)) = Res_{C}(γ) + Ram_{C}(γ).

Globalizing this construction over M_{g}, we are lead to consider the effective divisor

ℜ𝔗_{g} := {[C, x_{1}, …, x_{g}] ∈ M_{g,g−1} : ∃p ∈ C with H^{0}(C, K_{C}(−x_{1} −⋯−x_{g−1} −2p)) ≠ 0}.

The key ingredient in the proof of Theorem 0.2 is the calculation of the class of \overline{ℜ𝔗}_{g}:

Theorem 0.4. The closure in \overline{M}_{g,g−1} of the locus ℜ𝔗_{g} := \{[C, x_{1}, …, x_{g−1}] ∈ M_{g,g−1} : x_{1} + ⋯ + x_{g−1} ∈ Res_{C}(γ)} is linearly equivalent to,

\overline{ℜ𝔗}_{g} ≡ −4(g − 7)λ + 4(g − 2) ∑_{i=1}^{g−1} ψ_{i} − 2δ_{irr} − (12g − 22)δ_{0:2}−

− ∑_{i=0}^{g} ∑_{s=0}^{i−1}(2i^{3} − 5i^{2} − 3i + 4g − 4i^{2}s + 14si − 6gs − s + 2s^{2}g − 3s^{2} + 2)δ_{i:s} ∈ Pic(\overline{M}_{g,g−1}).

In particular we note that condition (3) is satisfied. Since by construction, \overline{ℜ𝔗}_{g} is 𝔖_{g−1}-invariant, it descends to an effective divisor ℜ̃𝔗_{g} on \overline{C}_{g,g−1} which, as it turns out, spans an extremal ray of the cone Eff(\overline{C}_{g,g−1}). Indeed, the universal theta divisor comes equipped with the rational involution τ : \overline{C}_{g,g−1} −−→ \overline{C}_{g,g−1} given by

τ([C, x_{1} + ⋯ + x_{g−1}]) := [C, y_{1} + ⋯ + y_{g−1}],

where O_{C}(y_{1} + ⋯ + y_{g−1} + x_{1} + ⋯ + x_{g−1}) = K_{C}. Then ℜ̃𝔗_{g} is the pull-back of the boundary divisor Δ̃_{0:2} ⊂ \overline{C}_{g,g−1} under this map. Since the extremality of Δ̃_{0:2} is easy to establish, the following result comes naturally:

Theorem 0.5. The effective divisor ℜ̃𝔗_{g} is covered by irreducible curves Γ_{g} ⊂ \overline{C}_{g,g−1} such that Γ_{g} · ℜ̃𝔗_{g} < 0. In particular ℜ̃𝔗_{g} ∈ Eff(\overline{C}_{g,g−1}) is a non-movable extremal effective divisor.

The curves Γ_{g} have a simple modular construction. One fixes a general linear series A ∈ W_{g+1}^{2}(C), in particular A is complete and has only ordinary ramification points. The general point of Γ_{g} corresponds to an element [C, D] ∈ \overline{C}_{g,g−1}, where D ∈ C_{g−1} is an effective divisor such that H^{0}(C, A ⊗ O_{C}(−2p − D)) ≠ 0, for some point p ∈ C, that is, D is the residual divisor cut out by a tangent line to the degree g +1 plane model of C given by A. Once more we refer to Section 2 for details.

We explain briefly how Theorem 0.4 implies the statement about the Kodaira dimension of \overline{C}_{g,g−1}. We choose an effective divisor D ≡ aλ−∑_{i=0}^{[g/2]} b_{i}δ_{i} ∈ Eff(\overline{M}_{g}) on the moduli space of curves, with a, b_{i} ≥ 0, having slope s(D) := \frac{a}{min_{i}b_{i}} as small as possible. Then note that the following linear combination

π^{∗}(K_{\overline{C}_{g,g−1}})−\frac{1}{6g − 11}(\frac{3}{2}[\overline{ℜ𝔗}_{g}]−(12g−25)ϕ^{∗}(D)−∑_{i=1}^{g−1} ψ_{i}−((84g−185)−(12g−25)s(D))λ)

is expressible as a positive combination of boundary divisors on \overline{M}_{g,g−1}. Since, as already pointed out, the class ∑_{i=1}^{g−1} ψ_{i} ∈ Pic(\overline{M}_{g,g−1}) descends to a big class on \overline{C}_{g,g−1}, one obtains the following:

Corollary 0.6. For all g such that the slope of the moduli space of curves satisfies the inequality

s(\overline{M}_{g}) := inf_{D∈Eff(\overline{M}_{g})}s(D) < \frac{84g − 185}{12g − 25},

the universal theta divisor 𝔗𝔥_{g} is of general type.

The bound appearing in Corollary 0.6 holds precisely when g ≥ 12; for g such that g + 1 is composite, the inequality s(\overline{M}_{g}) ≤ 6 + 12/(g + 1) is well-known, and D can be chosen to be a Brill-Noether divisor \overline{M}_{g,d}^{r} corresponding to curves with a 𝔤_{d}^{r} when the Brill-Noether number ρ(g, r, d) = −1, cf. [EH1]. When g + 1 is prime and g ≠ 12, then in practice g = 2k − 2 ≥ 16, and D can be chosen to be the Gieseker-Petri \overline{GP}_{g,k}^{1} consisting of curves C possessing a pencil A ∈ W_{k}^{1}(C) such that the Petri map μ_{0}(C, A) : H^{0}(C, A) ⊗ H^{0}(C, K_{C} ⊗ A^{∨}) → H^{0}(C, K_{C}) is not an isomorphism. When g = 12, one has to use the divisor constructed on \overline{M}_{12} in [FV1]. Finally, when g ≤ 11 it is known that s(\overline{M}_{g}) ≥ 6 + 12/(g + 1) and inequality (0.6) is not satisfied. In fact, as already pointed out κ(𝔗𝔥_{g}) = −∞ in this range.

The proof of Theorem 0.3 proceeds along similar lines, and relies on finding an explicit 𝔖_{g−2}-invariant extremal ray of the cone of effective divisors on \overline{M}_{g,g−2}. A representative of this ray is characterized by the geometric condition that the marked points appear in the same fibre of a pencil of degree g − 1. One can construct such divisors on all moduli spaces \overline{M}_{g,n} with 1 ≤ n ≤ g − 2, cf. Section 3.

Theorem 0.7. The closure inside \overline{M}_{g,g−2} of the locus

F_{g,1} := {[C, x_{1}, …, x_{g−2}] ∈ M_{g,g−2} : ∃A ∈ W_{g−1}^{1}(C) with H^{0}(C, A(− ∑_{i=1}^{g−2} x_{i})) ≠ 0}

is a non-movable, extremal effective divisor on \overline{M}_{g,g−2}. Its class is given by the formula:

\overline{F}_{g,1} ≡ −(g −12)λ+(g −3) ∑_{i=1}^{g−2} ψ_{i} −δ_{irr} − \frac{1}{2} ∑_{s=2}^{g−2} s(g −4+sg −2s) δ_{0:s} −⋯ ∈ Pic(\overline{M}_{g,g−2}).

Note that again, inequality (3) is satisfied, hence \overline{F}_{g,1} can be used to prove that K_{\overline{C}_{g,g−2}} is big. Moreover, \overline{F}_{g,1} descends to an extremal divisor F̃_{g,1} ∈ Eff(\overline{C}_{g,g−2}). In

fact, we shall show that F̃_{g,1} is swept by curves intersecting its class negatively.

Divisors similar to those considered in Theorems 0.4 and 0.7 can be constructed on other moduli spaces. On \overline{M}_{g,g−3} we construct an extremal divisor using a somewhat similar construction. If D ∈ C_{g−3} is a general effective divisor of degree g −3 on a curve

[C] ∈ M_{g}, we observe that K_{C} ⊗ O_{C}(−D) ∈ W_{g+1}^{2}(C). A natural codimension one condition on \overline{M}_{g,g−3} is that this plane model have a triple point (a similar construction requiring instead that K_{C} ⊗ O_{C}(−D) have a cusp, produces a ”less extremal” divisor):

Theorem 0.8. The closure inside \overline{M}_{g,g−3} of the locus

D_{g} := {[C, x_{1}, …, x_{g−3}] ∈ M_{g,g−3} : ∃L ∈ W_{g}^{2}(C) with H^{0}(C, L(− ∑_{i=1}^{g−3} x_{i})) ≠ 0}

is an effective divisor. Its class in Pic(\overline{M}_{g,g−3}) is equal to

\overline{D}_{g} ≡ −\frac{2(g − 17)}{3}\binom{g − 3}{2}λ+\frac{2g − 3}{3}\binom{g − 4}{2} ∑_{i=1}^{g−3} ψ_{i}−\binom{g − 3}{2}δ_{irr}−(g^{2}−5g+5)(g−5)δ_{0:2}−· · · .

1. Cones of divisors on universal symmetric products

The aim of this section is to establish certain facts about boundary divisors on \overline{M}_{g,n} and \overline{C}_{g,n}, see [AC] for a standard reference. We follow the convention set in [FV2], that is, if M is a Deligne-Mumford stack, we denote by M its coarse moduli space.

For an integer 0 ≤ i ≤ [g/2] and a subset T ⊂ {1, …, n}, we denote by Δ_{i:T} the closure in \overline{M}_{g,n} of the locus of n-pointed curves [C_{1} ∪ C_{2}, x_{1}, …, x_{n}], where C_{1} and C_{2} are smooth curves of genera i and g − i respectively meeting transversally in one point, and the marked points lying on C_{1} are precisely those indexed by T. We define δ_{i:T} := [Δ_{i:T}]_{ℚ} ∈ Pic(\overline{M}_{g,n}). For 0 ≤ i ≤ [g/2] and 0 ≤ s ≤ g, we set

Δ_{i:s} := ∑_{#(T)=s} δ_{i:T} , δ_{i:s} := [Δ_{i:s}]_{ℚ} ∈ Pic(\overline{M}_{g,n}).

By convention, δ_{0:s} := ∅, for s < 2, and δ_{i:s} := δ_{g−i:n−s}. If ϕ : \overline{M}_{g,n} → \overline{M}_{g} is the morphism forgetting the marked points, we set λ := ϕ^{∗}(λ) and δ_{irr} := ϕ^{∗}(δ_{irr}), where δ_{irr} := [Δ_{irr}] ∈ Pic(\overline{M}_{g}) denotes the class of the locus of irreducible nodal curves. Furthermore, ψ_{1}, …, ψ_{n} ∈ Pic(\overline{M}_{g,n}) are the cotangent classes corresponding to the marked points. The canonical class of \overline{M}_{g,n} is computed via Kodaira-Spencer theory:

(4)  K_{\overline{M}_{g,n}} ≡ 13λ − 2δ_{irr} + ∑_{i=1}^{n} ψ_{i} − 2 ∑_{{T ⊂\{1,...,n\} \atop i≥0}} δ_{i:T} − δ_{1:∅} ∈ Pic(\overline{M}_{g,n}).

Let \overline{C}_{g,n} := \overline{M}_{g,n}/𝔖_{n} be the universal symmetric product and π : \overline{M}_{g,n} → \overline{C}_{g,n} (respectively φ : \overline{C}_{g,n} → \overline{M}_{g}) the projection (respectively the forgetful map), so that ϕ = φ ∘ π. We denote by λ̃, δ̃_{irr}, δ̃_{i:c} := [Δ̃_{i:c}] ∈ Pic(\overline{C}_{g,n}) the divisor classes on the symmetric product pulling-back to the same symbols on \overline{M}_{g,n}. Clearly, π^{∗}(λ̃) = λ, π^{∗}(δ̃_{irr}) = δ_{irr}, π^{∗}(δ̃_{i:c}) = δ_{i:c}; in the case i = 0, c = 2, this reflects the branching of the map π along the divisor Δ̃_{0:2} ⊂ \overline{C}_{g,n}. Following [FV2], let 𝕃 denote the line bundle on \overline{C}_{g,n}, having fibre

𝕃[C, x_{1} + ⋯ + x_{n}] := T_{x_{1}}^{∨}(C) ⊗ ⋯ ⊗ T_{x_{n}}^{∨}(C),

over a point [C, x_{1} + ⋯ + x_{n}] := π([C, x_{1}, …, x_{n}]) ∈ \overline{C}_{g,n}. We set ψ̃:= c_{1}(𝕃), and note:

(5)  π^{∗}(ψ̃) = ∑_{i=1}^{n}(ψ_{i} − ∑_{i∈T ⊂\{1,...,n\}} δ_{0:T} ) = ∑_{i=1}^{n} ψ_{i} − ∑_{s=2}^{n} s δ_{0:s} ∈ Pic(\overline{M}_{g,n}).

Proposition 1.1. For g ≥ 3 and n ≥ 0, the morphism π^{∗} : Pic(\overline{C}_{g,n})_{ℚ} → Pic(\overline{M}_{g,n})_{ℚ} is injective. Furthermore, there is an isomorphism of groups Pic(\overline{C}_{g,n})_{ℚ} →^{≅} N^{1}(\overline{C}_{g,n})_{ℚ}.

Proof. The first assertion is an immediate consequence of the existence of the norm morphism Nm_{π} : Pic(\overline{M}_{g,n}) → Pic(\overline{C}_{g,n}), such that Nm_{π}(π^{∗}(L)) = L^{⊗deg(π)}, for every L ∈ Pic(\overline{C}_{g,n}). The second part comes from the isomorphism Pic(\overline{M}_{g,n})_{ℚ} →^{≅} N^{1}(\overline{M}_{g,n})_{ℚ}, coupled with the commutativity of the obvious diagrams relating the Picard and Néron-Severi groups of \overline{M}_{g,n} and \overline{C}_{g,n} respectively. □

One may thus identify Pic(\overline{C}_{g,n})_{ℚ} ≅ Pic(\overline{M}_{g,n})_{ℚ}^{𝔖_{n}}. The Riemann-Hurwitz formula applied to the branched covering π : \overline{M}_{g,n} → \overline{C}_{g,n} yields,

π^{∗}(K_{\overline{C}_{g},n}) = K_{\overline{M}_{g,n}} − δ_{0:2} ≡ 13λ + ∑_{i=1}^{n} ψ_{i} − 2δ_{irr} − 3δ_{0:2} − 2 ∑_{s=3}^{n} δ_{0:s} − · · · .

As expected, the sum of cotangent classes descends to a big line bundle on \overline{C}_{g,n}.

Proposition 1.2. The divisor class N_{g,n} := ψ̃ + ∑_{s=2}^{n} sδ̃_{0:s} ∈ Eff(\overline{C}_{g,n}) is big and nef.

Proof. The class N_{g,n} is characterized by the property that π^{∗}(N_{g,n}) = ∑_{i=1}^{n} ψ_{i}. This is a nef class on \overline{M}_{g,n}, in particular, N_{g,n} is nef on \overline{C}_{g,n}. To establish that N_{g,n} is big, we express it as a combination of effective classes and the class κ̃_{1} ∈ Pic(\overline{C}_{g,n}), where

π^{∗}(κ̃_{1}) = κ_{1} = 12λ + ∑_{i=1}^{n} ψ_{i} − δ_{irr} − ∑_{i=0}^{[g/2]} ∑_{s≥0} δ_{i:s} ∈ Pic(\overline{M}_{g,n}).

Since π^{∗}(κ̃_{1}) is ample on \overline{M}_{g,n}, it follows that κ̃_{1} is ample as well. To finish the proof, we exhibit a suitable effective class on \overline{M}_{g,n} having negative λ-coefficient. For that purpose, we choose W_{g,n} ⊂ C_{g,n} to be the locus of effective divisors having a Weierstrass point in their support. For i = 1, …, n, we denote by σ_{i} : \overline{M}_{g,n} → \overline{M}_{g,1} the morphism forgetting all but the i-th point, and let

\overline{W} ≡ −λ + \binom{g + 1}{2}ψ − ∑_{i=1}^{g−1} \binom{g − i + 1}{2}δ_{i:1} ∈ Eff(\overline{M}_{g,1}),

be the class of the divisor of Weierstrass points on the universal curve. Then one finds

π^{∗}(\overline{W}_{g,n}) ≡ ∑_{i=1}^{n} σ_{i}^{∗}(\overline{W}) = −nλ + \binom{g + 1}{2} ∑_{i=1}^{n} ψ_{i} − \binom{g + 1}{2} ∑_{s=2}^{n} sδ_{0:s} − ⋯ ∈ Pic(\overline{M}_{g,n}),

and \overline{W}_{g,n} ≡ −gλ̃ + \binom{g+1}{2}ψ̃ − ∑_{i=1}^{[g/2]} ∑_{s≥0} b_{i:s}δ̃_{i:s}, where b_{i:s} > 0. One checks that N_{g,n} can be written as a ℚ-combination with positive coefficients of the ample class κ̃_{1}, the effective class [\overline{W}_{g,n}] and other boundary divisor classes. In particular, N_{g,n} is big. □

2. The universal ramification locus of the Gauss map

We begin the calculation of the divisor \overline{ℜ𝔗}_{g}, and for a start we consider its restriction ℜ𝔗_{g} to M_{g,g−1}. Recall that ℜ𝔗_{g} is defined as the closure of the locus of pointed curves [C, x_{1}, …, x_{g−1}] ∈ M_{g,g−1}, such that there exists a holomorphic form on C vanishing at x_{1}, …, x_{g−1} and having an unspecified double zero.

Let u : M_{g,g−1}^{(1)} → M_{g,g−1} be the universal curve over the stack of (g − 1)-pointed smooth curves and we denote by ([C, x_{1}, …, x_{g−1}], p) ∈ M_{g,g−1}^{(1)} a general point, where [C, x_{1}, …, x_{g−1}] ∈ M_{g,g−1} and p ∈ C is an arbitrary point. For i = 1, …, g − 1, let Δ_{ip} ⊂ M_{g,g−1}^{(1)} be the diagonal divisor given by the equation p = x_{i}. Furthermore, for i = 1, …, g − 1 we consider as before the projections σ_{i} : M_{g,g−1}^{(1)} → M_{g,1} (respectively σ_{p} : M_{g,g−1}^{(1)} → M_{g,1}), obtained by forgetting all marked points except x_{i} (respectively p), and then set K_{i} := σ_{i}^{∗}(ω_{ϕ}) ∈ Pic(M_{g,g−1}^{(1)}) and K_{p} := σ_{p}^{∗}(ω_{ϕ}) ∈ Pic(M_{g,g−1}^{(1)}). We consider the following cartesian diagram of stacks

X ⟶^{q} M_{g,g−1}^{(1)}
↓^{f}  ↓
M_{g,1} ⟶^{ϕ} M_{g}

in which all the morphisms are smooth and ϕ (hence also q) is proper. For 1 ≤ i ≤ g − 1 there are tautological sections r_{i} : M_{g,g−1}^{(1)} → X as well as r_{p} : M_{g,g−1}^{(1)} → X, and set E_{i} := Im(r_{i}), E_{p} := Im(r_{p}). Thus {E_{i}}_{i=1}^{g−1} and E_{p} are relative divisors over q.

For a point ([C, x_{1}, …, x_{g−1}], p) ∈ M_{g,g−1}^{(1)}, we denote D := ∑_{i=1}^{g−1} x_{i} + 2p ∈ C_{g+1}, and have the following exact sequence:

0 → \frac{H^{0}(O_{C}(D))}{H^{0}(O_{C})} → H^{0}(O_{D}(D)) →^{α_{D}} H^{1}(O_{C}) → H^{1}(O_{C}(D)) → 0.

In particular, the morphisms α_{D} globalize to a morphism of vector bundles over M_{g,g−1}^{(1)}

α : A := q_{∗}(O_{X} (∑_{i=1}^{g−1} E_{i} + 2E_{p})/O_{X} ) → R^{1}q_{∗}O_{X} .

The subvariety Z := {([C, x_{1}, …, x_{g−1}], p) ∈ M_{g,g−1}^{(1)} : H^{0}(K_{C}(−2p − ∑_{i=1}^{g−1} x_{i})) ≠ 0} is the non-surjectivity locus of α and ℜ𝔗_{g} := u_{∗}(Z) ⊂ M_{g,g−1}. The class of Z is equal to

[Z] = c_{2}(A^{∨} − (R^{1}q_{∗}O_{X} )^{∨}) = c_{2}(−q_{!}O_{X} (∑_{i=1}^{g−1} E_{i} + 2E_{p})) ∈ A^{2}(M_{g,g−1}^{(1)}),

where the last term can be computed by Grothendieck-Riemann-Roch:

ch(q_{!}O_{X} (∑_{i=1}^{g−1} E_{i} + 2E_{p})) = q_{∗}[(∑_{k≥0} \frac{(∑_{i=1}^{g−1} E_{i} + 2E_{p})^{k}}{k!} ) · (1 − \frac{c_{1}(ω_{q})}{2} + \frac{c_{1}^{2}(ω_{q})}{12} + ⋯)],

and we are interested in evaluating the terms of degree 1 and 2 in this expression. The result of applying GRR to the morphism q, can be summarized as follows:

Lemma 2.1. One has the following relations in A^{∗}(M_{g,g−1}^{(1)}): (i)

ch_{1}(q_{∗}(O_{X} (∑_{i=1}^{g−1} E_{i} + 2E_{p}))) = λ − ∑_{i=1}^{g−1} K_{i} − 3K_{p} + 2 ∑_{i=1}^{g−1} Δ_{ip}.

(ii)

ch_{2}(q_{∗}(O_{X} (∑_{i=1}^{g−1} E_{i} + 2E_{p}))) = \frac{5}{2}K_{p}^{2} + \frac{1}{2} ∑_{i=1}^{g−1} K_{i}^{2} − 2 ∑_{i=1}^{g−1}(K_{i} + K_{p}) · Δ_{ip}.

Proof. We apply systematically the push-pull formula and the following identities:

E_{i}^{2} = −E_{i} ·q^{∗}(K_{i}), E_{p}^{2} = −E_{p} ·q^{∗}(K_{p}), E_{i} ·c_{1}(ω_{q}) = E_{i} ·q^{∗}(K_{i}), E_{p} ·c_{1}(ω_{q}) = E_{p} ·q^{∗}(K_{p}),

E_{i} · E_{j} = 0 for i ≠ j, E_{i} · E_{p} = E_{i} · q^{∗}(Δ_{ip}), and q_{∗}(c_{1}^{2}(ω_{q})) = 12λ.

□

Proposition 2.2. The formula ℜ𝔗_{g} ≡ −4(g − 7)λ + (4g − 8) ∑_{i=1}^{g−1} ψ_{i} ∈ Pic(M_{g,g−1}) holds.

Proof. We apply the results of Lemma 2.1, as well as the formulas from [HM] p. 55, in order to estimate the push-forward under u of the degree 2 monomials in tautological classes. Setting F := q_{∗}(O_{X} (∑_{i=1}^{g−1} E_{i} + 2E_{p})), we obtain that

u_{∗}(ch_{1}^{2}(F)) = −(8g − 116)λ + (8g − 24) ∑_{i=1}^{g−1} ψ_{i}, and u_{∗}(ch_{2}(F)) = 30λ − 4 ∑_{i=1}^{g−1} ψ_{i},

hence [ℜ𝔗_{g}] = u_{∗}(ch_{1}^{2}(F) − 2ch_{2}(F))/2, and the claimed formula follows at once. □

We proceed now towards proving Theorem 0.4 and expand the divisor class [\overline{ℜ𝔗}_{g}] ∈ Pic(\overline{M}_{g,g−1}) in the standard basis of the Picard group, that is,

\overline{ℜ𝔗}_{g} ≡ aλ + c ∑_{i=1}^{g−1} ψ_{i} − b_{irr}δ_{irr} − ∑_{i=0}^{g} ∑_{s=0}^{i−1} b_{i:s}δ_{i:s}.

We have just computed a = −4(g − 7) and c = 4(g − 2). The remaining coefficients are determined by intersecting \overline{ℜ𝔗}_{g} with curves lying in the boundary of \overline{M}_{g,g−1} and understanding how \overline{ℜ𝔗}_{g} degenerates. We begin with the coefficient b_{0:2}:

Proposition 2.3. One has the relation (4g − 6)c − (g − 2)b_{0:2} = (4g − 2)(g − 2). It follows that b_{0:2} = 12g − 22.

Proof. We fix a general pointed curve [C, x_{1}, …, x_{g−2}] ∈ M_{g,g−2} and consider the family

C_{x_{g−1}} := \{[C, x_{1}, …, x_{g−2}, x_{g−1}] : x_{g−1} ∈ C\} ⊂ \overline{M}_{g,g−1}.

The curve C_{x_{g−1}} is the fibre over [C, x_{1}, …, x_{g−2}] of the morphism \overline{M}_{g,g−1} → \overline{M}_{g,g−2} forgetting the point labeled by x_{g−1}. Note that C_{x_{g−1}} · ψ_{i} = 1 for i = 1, …, g − 2 and C_{x_{g−1}} · ψ_{g−1} = 3g − 4 = 2g − 2 + (g − 2). Obviously C_{x_{i}} · δ_{0:2} = g − 2 and the points in the intersection correspond to the case when x_{g−1} collides with one of the fixed points x_{1}, …, x_{g−2}. The intersection of C_{x_{i}} with the remaining generators of Pic(\overline{M}_{g,g−1}) is equal to zero. We set A := K_{C} ⊗ O_{C}(−x_{1} − ⋯ − x_{g−2}) ∈ W_{g}^{1}(C). By the generality assumption, h^{0}(C, A) = 2, and all ramification points of A are simple. Pointed curves in the intersection C_{x_{g−1}} · \overline{ℜ𝔗}_{g} correspond to points x_{g−1} ∈ C, such that there exists a (ramification) point p ∈ C with H^{0}(C, A ⊗ O_{C}(−2p − x_{g−1})) ≠ 0. The pencil A carries 4g − 2 ramification points. For each of them there are g − 2 possibilities of choosing x_{g−1} ∈ C in the same fibre as the ramification point, hence the conclusion follows. □

Next we determine the coefficient b_{irr}. First we note that the relation

(6)  a − 12b_{irr} + b_{1:0} = 0

holds. Indeed, the divisor \overline{ℜ𝔗}_{g} is disjoint from the curve in Δ_{1:0} ⊂ \overline{M}_{g,g−1}, obtained from a fixed pointed curve [C, x_{1}, …, x_{g−1}, q] ∈ \overline{M}_{g−1,g}, by attaching at the point q a pencil of plane cubics along a section of the pencil induced by one of the 9 base points.

Proposition 2.4. One has the relation b_{irr} = 2.

Proof. We fix a general curve [C, q, x_{1}, …, x_{g−1}] ∈ \overline{M}_{g−1,g}, and we define the family

C_{irr} := \{[C/t ∼ q, x_{1}, …, x_{g−1}] : t ∈ C\} ⊂ Δ_{irr} ⊂ \overline{M}_{g,g−1}.

Then C_{irr} · ψ_{i} = 1 for i = 1, …, g − 1, C_{irr} · δ_{irr} = −(deg(K_{C}) + 2) = −2g + 2, and finally C_{irr} ·δ_{1:0} = 1. All other intersection numbers with generators of Pic(\overline{M}_{g,g−1}) equal zero.

We fix an effective divisor D ∈ C_{e} of degree e ≥ g (for instance D = q + ∑_{i=1}^{g−1} x_{i}). For each pair of points (t, p) ∈ C × C, there is an exact sequence on C

0 → H^{0}(C, K_{C}(q + t − 2p − ∑_{i=1}^{g−1} x_{i})) → H^{0}(C, K_{C}(D + q + t − 2p − ∑_{i=1}^{g−1} x_{i})) →^{β_{t,p}}

H^{0}(D, K_{C}(D + q + t − 2p − ∑_{i=1}^{g−1} x_{i})) → H^{1}(C, K_{C}(q + t − 2p − ∑_{i=1}^{g−1} x_{i})) → 0.

The intersection C_{irr} · \overline{ℜ𝔗}_{g} corresponds to the locus of pairs (t, p) ∈ C × C such that the map β_{t,p} is not injective. On the triple product of C, we consider two of the projections f : C×C×C → C×C and p_{1} : C×C×C → C given by f(x, t, p) = (t, p) and p_{1}(x, t, p) = x, then set A := K_{C}(q − ∑_{i=1}^{g−1} x_{i}) ∈ Pic^{g−2}(C). We denote by Δ_{12}, Δ_{13} ⊂ C × C × C the corresponding diagonals, and finally, introduce the line bundle on C × C × C

F := p_{1}^{∗}(A) ⊗ O_{C×C×C}(Δ_{12} − 2Δ_{13}).

Applying the Porteous formula, one can write

C_{irr} · \overline{ℜ𝔗}_{g} = c_{2}(R^{1}f_{∗}F − R^{0}f_{∗}F) = \frac{ch_{1}^{2}(f_{!}F) + 2ch_{2}(f_{!}F)}{2} ∈ A^{2}(C × C).

We evaluate ch_{i}(f_{!}F) using GRR applied to the morphism f, that is,

ch(f_{!}F) = f_{∗}[(∑_{a≥0} \frac{(p_{1}^{∗}(A) + Δ_{12} − 2Δ_{13})^{a}}{a!} ) · (1 − \frac{1}{2}p_{1}^{∗}(K_{C}))].

Denoting by F_{1}, F_{2} ∈ H^{2}(C × C) the class of the fibres, after calculations one finds that

ch_{1}(f_{∗}F) = −(g − 2)F_{1} − 4(g − 2)F_{2} − 2Δ_{C} ∈ H^{2}(C × C, ℚ),
ch_{2}(f^{∗}F) = −2(g − 2) ∈ H^{4}(C × C, ℚ),

that is, c_{2}(R^{1}f_{∗}F − R^{0}f_{∗}F) = 4(g − 2)(g − 1). Coupled with (6), this yields b_{irr} = 2. □

We are left with the task of determining the coefficient of δ_{i:s} in the expansion of [\overline{ℜ𝔗}_{g}]. This requires solving a number of enumerative geometry problems in the spirit of de Jonquières’ formula. We fix integers 0 ≤ i ≤ g and s ≤ i − 1 as well as general pointed curves [C, x_{1}, …, x_{s}] ∈ \overline{M}_{i,s} and [D, q, x_{s+1}, …, x_{g−1}] ∈ \overline{M}_{g−i,g−s}, then construct a pencil of stable curves of genus g, by identifying the fixed point q ∈ D with a variable point, also denoted by q, on the component C:

C_{i:s} := \{[C ∪_{q} D, x_{1}, …, x_{s}, x_{s+1}, …, x_{g−1}] : q ∈ C\} ⊂ Δ_{i:s} ⊂ \overline{M}_{g,g−1}.

We summarize the non-zero intersection numbers of C_{i:s} with generators of Pic(\overline{M}_{g,g−1}):

C_{i:s} · ψ_{1} = ⋯ = C_{i:s} · ψ_{s} = 1, C_{i:s} · δ_{i:s−1} = i, C_{i:s} · δ_{i:s} = 2i − 2 + s.

Theorem 2.5. We fix integers 0 ≤ i ≤ g and 0 ≤ s ≤ i − 1. Then, the following formula holds:

b_{i:s} = 2i^{3} − 5i^{2} − 3i + 4g − 4i^{2}s + 14si − 6gs − s + 2s^{2}g − 3s^{2} + 2.

In the proof an essential role is played by the following calculation:

Proposition 2.6. Let i, s be integers such that 0 ≤ s ≤ i − 1, and [C, x_{1}, …, x_{s}] ∈ M_{i,s} a general pointed curve. The number of pairs (q, p) ∈ C × C such that

H^{0}(C, K_{C} ⊗ O_{C}(−x_{1} − ⋯ − x_{s} − (i − s − 1)q − 2p)) ≠ 0,

is equal to a(i, s) := 2(i − s − 1)(2i^{3} − 5i^{2} + i + 2 − 2i^{2}s + 3is).

Remark 2.7. By specializing, one recovers well-known formulas in enumerative geometry. For instance, a(3, 0) = 56 is twice the number of bitangents of a smooth plane quartic, whereas a(4, 0) = 324 equals the number of canonical divisors of type 3q + 2p + x ∈ |K_{C}|, where [C] ∈ M_{4}. This matches de Jonquières’ formula, cf. [ACGH] p.359.

Proof of Theorem 2.5. We fix a general point [C∪_{q}D, x_{1}, …, x_{g−1}] ∈ C_{i:s}·\overline{ℜ𝔗}_{g}, corresponding to a point q ∈ C. We shall show that q is not one of the marked points x_{1}, …, x_{s} on C, then give a geometric characterization of such points and count their number. Let

ω_{D} ∈ H^{0}(D, K_{D} ⊗ O_{D}(2iq)) and ω_{C} ∈ H^{0}(C, K_{C} ⊗ O_{C}(2g − 2i)q)

be the aspects of the section of the limit canonical series on C ∪_{q} D, which vanishes doubly at an unspecified point p ∈ C ∪ D as well as along the divisor x_{1} + ⋯ + x_{g−1}. The condition ord_{q}(ω_{C}) + ord_{q}(ω_{D}) ≥ 2g − 2, comes from the definition of a limit linear series. We distinguish two cases depending on the position of the point p. If p ∈ D then,

div(ω_{C}) ≥ x_{1} + ⋯ + x_{s}, div(ω_{D}) ≥ x_{s+1} + ⋯ + x_{g−1} + 2p.

Since the points q, x_{s+1}, …, x_{g−1} ∈ D are general, we find that ord_{q}(ω_{D}) ≤ i + s − 2. Moreover, K_{D} ⊗ O_{D}((i − s + 2)q − x_{s+1} − ⋯ − x_{g−1}) ∈ W_{g−i+1}^{1}(D) is a pencil, and p ∈ D is one of its (simple) ramification points. The Hurwitz formula gives 4(g − i) choices for such p ∈ D.

By compatibility, ord_{q}(ω_{C}) ≥ 2g − i − s. A parameter count implies that equality must hold. The condition H^{0}(C, K_{C} ⊗O_{C}(−x_{1} −⋯−x_{s} −(i−s)q) ≠ 0, is equivalent to asking that q ∈ C be a ramification point of K_{C} ⊗ O_{C}(− ∑_{j=1}^{s} x_{j}) ∈ W_{2i−2−s}^{i−s−1}(C). Since the points x_{1}, …, x_{s} ∈ C are chosen to be general, all ramification points of this linear series are simple and occur away from the marked points. From Plücker’s formula, the number of ramification points equals (i − s)(i^{2} − 1 − is). Multiplying this with the number of choices for p ∈ D, we obtain a total contribution of 4(g−i)(i−s)(i^{2}−is−1) to the intersection C_{i:s} · \overline{ℜ𝔗}_{g}, stemming from the case when p ∈ D. The proof that each of these points of intersection is to be counted with multiplicity 1 is standard and proceeds along the lines of [EH2] Lemma 3.4.

We assume now that p ∈ C. Keeping the notation from above, it follows that ord_{q}(ω_{D}) = i + s − 1 and ord_{q}(ω_{C}) = 2g − i − s − 1, therefore

0 ≠ σ_{C} ∈ H^{0}(C, K_{C} ⊗ O_{C}(− ∑_{j=1}^{s} x_{j} − (i − s − 1)q − 2p)).

The section ω_{D} is uniquely determined up to multiplication by scalars, whereas there are a(i, s) choices on the side of C, each counted with multiplicity 1.

In principle, the double zero of the limit holomorphic form could specialize to the point of attachment q ∈ C ∩ D, and we prove that this would contradict our generality hypothesis. One considers the semistable curve X := C ∪_{q_{1}} E ∪_{q_{2}} D, obtained from C ∪ D by inserting a smooth rational component E at q, where {q_{1}} := C ∩ E and {q_{2}} := D ∩ E. There also exist non-zero sections

ω_{D} ∈ H^{0}(D, K_{D}(2iq_{2})), ω_{E} ∈ H^{0}(E, O_{E}(2g − 2)), ω_{C} ∈ H^{0}(C, K_{C}((2g − 2i)q_{1})),

satisfying ord_{q_{1}}(ω_{C}) + ord_{q_{1}}(ω_{E}) ≥ 2g − 2 and ord_{q_{2}}(ω_{E}) + ord_{q_{2}}(ω_{D}) ≥ 2g − 2. Furthermore, ω_{E} vanishes doubly at a point p ∈ {q_{1}, q_{2}}^{c}. Since ω_{C} (respectively ω_{D}) also vanishes along the divisor x_{1} + ⋯ + x_{s} (respectively x_{s+1} + ⋯ + x_{g−1}), it follows that ord_{q_{1}}(ω_{C}) ≤ 2g − i − s and ord_{q_{2}}(ω_{D}) ≤ i + s − 1, hence by compatibility, ord_{q_{1}}(ω_{E}) + ord_{q_{2}}(ω_{E}) ≥ 2g − 3. This rules out the possibility of a further double zero and shows that this case does not occur.

To summarize, keeping in mind that the ψ-coefficient of [\overline{ℜ𝔗}_{g}] is equal to 4g − 8, we find the relation

(7) (2i − 2 + s)b_{i:s} − sb_{i:s−1} + s(4g − 8) = 4(g − i)(s − 1)(si − 2i + 2) + a(i, s).

For s = 0, we have by convention b_{i:−1} = 0, which gives b_{i:0} = 2i^{3} − 5i^{2} − 3i + 4g + 1. By induction, we find using recursion (7) the claimed formula for b_{i:s}. □

As already explained, having calculated the class [\overline{ℜ𝔗}_{g}] ∈ Pic(\overline{M}_{g,g−1}) and using

known bound on the slope s(\overline{M}_{g}), one derives that 𝔗𝔥_{g} is of general type when g ≥ 12. We discuss the last cases in Theorem 0.1 and thus complete the birational classification of 𝔗𝔥_{g}: End of proof of Theorem 0.1. We noted in the Introduction that for g ≤ 9 the space 𝔗𝔥_{g} is unirational, being the image of a variety which is birational to a Grassmann bundle over the rational Mukai variety V_{g}^{g−1}. When g ∈ \{10, 11\}, the space \overline{M}_{g,g−1} is uniruled [FP]. This implies the uniruledness of 𝔗𝔥_{g} as well. □

3. The Kodaira dimension of \overline{C}_{g,n}

In this section we provide results concerning the Kodaira dimension of the symmetric product \overline{C}_{g,n}, where n ≤ g−2. There are two cases depending on the parity of the difference g − n. When g − n is even, we introduce a subvariety inside C_{g,n}, consisting of divisors D ∈ C_{n} which appear in a fibre of a pencil of degree (g + n)/2 on a curve

[C] ∈ M_{g}. We set integers g ≥ 1 and 1 ≤ m ≤ g/2, then consider the locus

F_{g,m} := {[C, x_{1}, …, x_{g−2m}] ∈ M_{g,g−2m} : ∃A ∈ W_{g−m}^{1}(C) with H^{0}(C, A(− ∑_{j=1}^{g−2m} x_{j})) ≠ 0}.

A parameter count shows that F_{g,m} is expected to be an effective divisor on \overline{M}_{g,g−2m}. We shall prove this, then compute the class of its closure in \overline{M}_{g,g−2m}.

Theorem 3.1. Fix integers g ≥ 1 and 1 ≤ m ≤ g/2, then set n := g − 2m and d := g − m. The class of the compactification inside \overline{M}_{g,g−2m} of the divisor F_{g,m} is given by the formula:

\overline{F}_{g,m} ≡ (\frac{10n}{g − 2}\binom{g − 2}{d − 1} − \frac{n}{g}\binom{g}{d})λ + \frac{n − 1}{g − 1}\binom{g − 1}{d − 1} ∑_{j=1}^{n} ψ_{j} − \frac{n}{g − 2}\binom{g − 2}{d − 1}δ_{irr}−

− ∑_{s=2}^{n} \frac{s(n^{2} − g + sgn − sn)}{2(g − 1)(g − d)}\binom{g − 1}{d}δ_{0:s} − ⋯ ∈ Pic(\overline{M}_{g,n}).

Proof. We fix a general curve [C] ∈ M_{g} and consider the incidence correspondence

Σ := {(D, A) ∈ C_{g−2m} × W_{g−m}^{1}(C) : H^{0}(C, A ⊗ O_{C}(−D)) ≠ 0},

together with the projection π_{1} : Σ → C_{g−2m}. It follows from [F1] Theorem 0.5, that Σ is pure of dimension g − 2m − 1(= ρ(g, 1, g − m) + 1). To conclude that \overline{F}_{g,m} is a divisor inside \overline{M}_{g,g−2m}, it suffices to show that the general fibre of the map π_{1} is finite, which implies that ϕ^{−1}([C]) ∩ \overline{F}_{g,m} is a divisor in ϕ^{−1}([C]); we also note that the fibre ϕ^{−1}([C]) is isomorphic to the n-th Fulton-Macpherson configuration space of C. We specialize to the case D = (g − 2m) · p, where p ∈ C. One needs to show that for a general curve [C] ∈ M_{g}, there exist finitely many pencils A ∈ W_{g−m}^{1}(C) with h^{0}(C, A ⊗ O_{C}(−(g − m)p)) ≥ 1, for some point p ∈ C. This follows from [HM] Theorem B, or alternatively, by letting C specialize to a flag curve consisting of a rational spine and g elliptic tails, in which case the point p specializes to a (g − 2m)-torsion points on one of the elliptic tails (in particular it can not specialize to a point on the spine). For each of these points, the pencils in question are in bijective correspondence to points in a transverse intersection of Schubert cycles in G(2, g−m+1). In particular their number is finite.

In order to compute the class [\overline{F}_{g,m}], we expand it in the usual basis of Pic(\overline{M}_{g,n})

\overline{F}_{g,m} ≡ aλ + c ∑_{i=1}^{g−2m} ψ_{i} − b_{irr}δ_{irr} − ∑_{i,s≥0} b_{i:s}δ_{i:s},

then note that the coefficients a, c and b_{irr} respectively, have been computed in [F2] Theorem 4.9. The coefficient b_{0:2} is determined by intersecting \overline{F}_{g,m} with a fibral curve

C_{x_{n}} := \{[C, x_{1}, …, x_{n−1}, x_{n}] : x_{n} ∈ C\} ⊂ \overline{M}_{g,n},

corresponding to a general (n − 1)-pointed curve [C, x_{1}, …, x_{n−1}] ∈ \overline{M}_{g,n−1}. By letting the points x_{1}, …, x_{n−1} ∈ C coalesce to a point q ∈ C, points in the intersection C_{x_{n}}·\overline{F}_{g,m} are in 1 : 1 correspondence with points x_{n} ∈ C, such that h^{0}(C, A(−(n − 1)q − x_{n})) ≥ 1. This number equals (g − 2m − 1)\binom{g}{m}, see [HM] Theorem A, that is,

(2g + 2n − 4)c − (n − 1)b_{0:2} = C_{x_{n}} · \overline{F}_{g,m} =

(m + 1) #\{A ∈ W_{g−m}^{1}(C) : h^{0}(C, A ⊗ O_{C}(−(g − 2m − 1)q)) ≥ 1\} = (g − 2m − 1)\binom{g}{m},

which determines b_{0:2}. The coefficients b_{0:s} are computed recursively, by exhibiting an explicit test curve Γ_{0:s} ⊂ Δ_{0:s} which is disjoint from \overline{F}_{g,m}. We fix a general element [C, q, x_{s+1}, …, x_{n}] ∈ \overline{M}_{g,n+1−s} and a general s-pointed rational curve [P^{1}, x_{1}, …, x_{s}] ∈ \overline{M}_{0,s}. We glue these curves along a moving point q lying on the rational component:

Γ_{0:s} := \{[P^{1} ∪_{q} C, x_{1}, …, x_{s}, x_{s+1}, …, x_{n}] : q ∈ P^{1}\} ⊂ Δ_{0:s} ⊂ \overline{M}_{g,n}.

Clearly, Γ_{0:s} · \overline{F}_{g,m} = s c − (s − 2) b_{0:s} + s b_{0:s−1}. We claim Γ_{0:s} ∩ \overline{F}_{g,m} = ∅. Assume that on the contrary, one can find a point q ∈ P^{1} and a limit linear series 𝔤_{d}^{1} on P^{1} ∪_{q} C,

l = ((A, V_{C}), (O_{P}_{1}(d), V_{P}_{1})) ∈ G_{d}^{1}(C) × G_{d}^{1}(P^{1}),

together with sections σ_{C} ∈ V_{C} and σ_{P}_{1} ∈ V_{P}_{1}, satisfying ord_{q}(σ_{C}) + ord_{q}(σ_{P}_{1}) ≥ d and

div(σ_{C}) ≥ x_{s+1} + ⋯ + x_{n}, div(σ_{P}_{1}) ≥ x_{1} + ⋯ + x_{s}.

Since σ_{P}_{1} ≠ 0, one finds that ord_{q}(σ_{P}_{1}) ≤ g−m−s, hence by compatibility, ord_{q}(σ_{C}) ≥ s. We claim that this is impossible, that is, H^{0}(C, A⊗O_{C}(−sq−x_{1}−⋯−x_{n})) ≠ 0, for every A ∈ W_{g−m}^{1}(C). Indeed, by letting all points x_{s+1}, …, x_{n}, q ∈ C coalesce, the statement H^{0}(C, A ⊗ O_{C}(−(g − 2m) · q)) = 0, for a general [C, q] ∈ \overline{M}_{g,1} is a consequence of the ”pointed” Brill-Noether theorem as proved in [EH1] Theorem 1.1. This shows that

0 = Γ_{0:s} · \overline{F}_{g,m} = sc + (s − 2)b_{0:s} − sb_{0:s−1},

for 3 ≤ s ≤ n, which determines recursively all coefficients b_{0:s}. The remaining coefficients b_{i:s} with 1 ≤ i ≤ [g/2] can be determined via similar test curve calculations, but we skip these details. □

Keeping the notation from the proof of Theorem 3.1, a direct consequence is the calculation of the class of the divisor F_{g,m}[C] := π_{1}(Σ) inside C_{g−2m}. This offers an alternative proof of [Mus] Proposition III; furthermore the proof of Theorem 3.1, answers in the affirmative the question raised in loc.cit., concerning whether the cycle F_{g,m}[C] has expected dimension, and thus, it is a divisor on C_{g−2m}.

We denote by θ ∈ H^{2}(C_{g−2m}, ℚ) the class of the pull-back of the theta divisor, and by x ∈ H^{2}(C_{g−2m}, ℚ) the class of the locus {p_{0} + D : D ∈ C_{g−2m−1}} of effective divisors containing a fixed point p_{0} ∈ C. For a very general curve [C] ∈ M_{g}, the group N^{1}(C_{g−2m})_{ℚ} is generated by x and θ, see [ACGH].

Let F̃_{g,m} be the effective divisor on \overline{C}_{g,g−2m} to which \overline{F}_{g,m} descends, that is, π^{∗}(F̃_{g,m}) = \overline{F}_{g,m}. The class of F̃_{g,m} is completely determined by Theorem 3.1.

Corollary 3.2. Let [C] ∈ M_{g} be a general curve. The cohomology class of the divisor

F_{g,m}[C] := {D ∈ C_{g−2m} : ∃A ∈ W_{g−m}^{1}(C) such that H^{0}(C, A ⊗ O_{C}(−D)) ≠ 0}

is equal to (1 − \frac{2m}{g})\binom{g}{m}(θ − \frac{g}{g−2m}x). In particular, the class θ − \frac{g}{g−2m}x ∈ N^{1}(C_{g−2m})_{ℚ} is effective.

Proof. Let u : C_{g−2m} −−→ \overline{C}_{g,g−2m} be the rational map given by

u(x_{1} + ⋯ + x_{g−2m}) = [C, x_{1} + ⋯ + x_{g−2m}].

Note that u is well-defined outside the codimension 2 locus of effective divisors with support of length at most g − 2m − 2. We have that u^{∗}(δ̃_{0:2}) = δ_{C}, where δ_{C} := [Δ_{C}]/2 is the reduced diagonal. Its class is given by the MacDonald formula, cf. [K1] Lemma 7:

δ_{C} ≡ −θ + (g + d − 1)x ≡ −θ + (2g − 2m − 1)x.

Furthermore, u^{∗}(ψ̃) ≡ θ + δ_{C} + (g − n − 1)x, see [K2] Proposition 2.7. Thus F_{g,m}[C] ≡ u^{∗}([F̃_{g,m}]), and the conclusion follows after some calculations. □

The divisor F̃_{g,m} is defined in terms of a correspondence between pencils and effective divisors on curves, and it is fibred in curves as follows: We fix a complete pencil A ∈ W_{g−m}^{1}(C) with only simple ramification points. The variety of secant divisors

V_{g−2m}^{1}(A) := {D ∈ C_{g−2m} : H^{0}(C, A ⊗ O_{C}(−D)) ≠ 0}

is a curve (see [F1]), disjoint from the indeterminacy locus of the rational map u : C_{g−2m} −−→ \overline{C}_{g,g−2m}. We set Γ_{g−2m}(A) := u(V_{g−2m}^{1}(A)) ⊂ \overline{C}_{g,g−2m}. By varying [C] ∈ M_{g} and A ∈ W_{g−m}^{1}(C), the curves Γ_{g−2m}(A) fill-up the divisor F̃_{g,m}. It is natural to test the extremality of F̃_{g,m} by computing the intersection number Γ_{g−2m}(A) · F̃_{g,m}. To state the next result in a unified form, we adopt the convention \binom{a}{b} := 0, whenever b < 0.

Proposition 3.3. For all integers 1 ≤ m < g/2, we have the formula:

Γ_{g−2m}(A) · F̃_{g,m} = (m − 1)\binom{g − m − 2}{m}\binom{g}{m}.

In particular, Γ_{g−2}(A) · F̃_{g,1} = 0, and the divisor F̃_{g,1} ∈ Eff(\overline{C}_{g,g−2}) is extremal.

Proof. This is an immediate application of Corollary 3.2. The class [V_{g−2m}^{1}(A)] can be computed using Porteous’ formula, see [ACGH] p.342:

[V_{g−2m}^{1}(A)] ≡ ∑_{j=0}^{g−2m−1} \binom{−m − 1}{j}\frac{x^{j} · θ^{g−2m−j−1}}{(g − 2m − 1 − j)!} ∈ H^{2(g−2m−1)}(C_{g−2m}, ℚ).

Using the push-pull formula, we write Γ_{g−2m}(A) · F̃_{g,m} = F_{g,m}[C] · [V_{g−2m}^{1}(A)], then estimate the product using the identity x^{k}θ^{g−2m−k} = g!/(2m+k)! ∈ H^{2(g−2m)}(C_{g−2m}, ℚ) for 0 ≤ k ≤ g − 2m. For m = 1, observe that Γ_{g−2}(A) · F̃_{g,1} = 0. Since the curves of type Γ_{g−2}(A) cover F̃_{g,1}, this implies that [F̃_{g,1}] ∈ Eff(\overline{C}_{g,g−2}) generates an extremal ray. □

We can use Theorem 3.1 to describe the birational type of \overline{C}_{g,n} when 12 ≤ g ≤ 21 and 1 ≤ n ≤ g − 2. We recall that when g ≤ 9, the space \overline{C}_{g,n} is uniruled for all values of n. The transition cases g = 10, 11, as well as the case of the universal Jacobian \overline{C}_{g,g}, are discussed in detail in [FV2]. Furthermore \overline{C}_{g,n} is uniruled when n ≥ g + 1; in this case the symmetric product C_{n} of any curve [C] ∈ M_{g} is birational to a P^{n−g}-bundle over the Jacobian Pic^{n}(C). Our main result is that, in the range described above, \overline{C}_{g,n} is of general type in all the cases when \overline{M}_{g,n} is known to be of general type, see [Log], [F2]. We note however that the divisors \overline{F}_{g,m} only carry one a certain distance towards a full solution. The classification of \overline{C}_{g,n} is complete only when n ∈ \{g − 1, g − 2, g\}.

Theorem 3.4. For integers g = 12, …, 21, the universal symmetric product \overline{C}_{g,n} is of general type for all f(g) ≤ n ≤ g − 1, where f(g) is described in the following table.

\begin{matrix}g & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 \\ f(g) & 10 & 11 & 10 & 10 & 9 & 9 & 9 & 7 & 6 & 4 \end{matrix}

Proof. The strategy described in the Introduction to prove that K_{\overline{C}_{g,g−1}} is big, applies to the other spaces \overline{C}_{g,n}, with 1 ≤ n ≤ g − 2 as well. To show that \overline{C}_{g,n} is of general type, it suffices to produce an effective class on \overline{C}_{g,n} which pulls back via π to aλ + c ∑_{i=1}^{n} ψ_{i} − b_{irr}δ_{irr} − ∑_{i,s} b_{i:s}δ_{i:s} ∈ Eff(\overline{M}_{g,n})^{𝔖_{n}}, such that the following conditions are fulfilled:

(8)  \frac{a + s(\overline{M}_{g})(2c − b_{irr})}{13c} < 1 and \frac{b_{0:2}}{3c} > 1.

When g − n is even, we write g − n = 2m, and for all entries in the table above one can express K_{\overline{C}_{g,n}} as a positive combination of ∑_{i=1}^{n} ψ_{i}, [\overline{F}_{g,m}], φ^{∗}(D), where D ∈ Eff(\overline{M}_{g}), and other boundary classes.

If g − n = 2m + 1 with m ∈ ℤ_{≥0}, for each integer 1 ≤ j ≤ n + 1, we denote by ϕ_{j} : \overline{M}_{g,n+1} → \overline{M}_{g,n} the projection forgetting the j-th marked point and consider the effective 𝔖_{n}-invariant effective ℚ-divisor on \overline{M}_{g,n}

E := \frac{1}{n + 1} ∑_{j=1}^{n+1}(ϕ_{j})_{∗}(\overline{F}_{g,m} · δ_{0:\{j,n+1\}}) ∈ Eff(\overline{M}_{g,n}).

Using Theorem 3.1 as well as elementary properties of push-forwards of tautological classes, K_{\overline{C}_{g,n}} is expressible as a positive ℚ-combination of boundaries, [E], a pull-back of an effective divisor on \overline{M}_{g}, and the big and nef class ∑_{i=1}^{n} ψ_{i} precisely in the cases appearing in the table. □

Remark 3.5. When g ∉ \{12, 16, 18\}, the bound s(\overline{M}_{g}) ≤ 6 + 12/(g + 1), emerging from the slope of the Brill-Noether divisors, has been used to verify (8). In the remaining cases, we employ the better bounds s(\overline{M}_{12}) = 4415/642 < 6 + 12/13 (see [FV1]), and s(\overline{M}_{16}) = 407/61 < 6 + 12/17 see [F2], coming from Koszul divisors on \overline{M}_{12} and \overline{M}_{16} respectively. On \overline{M}_{18}, we use the estimate s(\overline{M}_{18}) ≤ 302/45 given by the class of the Petri divisor \overline{GP}_{18,10}^{1}, see [EH1]. Improvements on the estimate on s(\overline{M}_{g}) in the other cases, will naturally translate in improvements in the statement of Theorem 3.4.

4. An effective divisor on \overline{M}_{g,g−3}

The aim of this section is to prove Theorem 0.8. We begin by solving the following enumerative question which comes up repeatedly in the process of computing [\overline{D}_{g}].

Theorem 4.1. Let [C, p] ∈ M_{g,1} be a general pointed curve of genus g and 0 ≤ γ ≤ g − 3 a fixed integer. Then there exist a finite number of pairs (L, x) ∈ W_{g}^{2}(C) × C such that

H^{0}(C, L ⊗ O_{C}(−γ x − (g − 3 − γ) p)) ≥ 1.

Their number is computed by the formula

N(g, γ) := \frac{g(g − 1)(g − 5)}{3}γ(γg − 3γ − 1).

Proof. We introduce auxiliary maps χ : C × C_{3} → C_{γ+3} and ι : C_{γ+3} → C_{g} given by,

χ(x, D) := γ · x + D, and ι(E) := E + (g − 3 − γ) · p.

The number we evaluate is N(g, γ) := χ^{∗}ι^{∗}([C_{g}^{2}]), where C_{g}^{2} := {D ∈ C_{g} : dim|D| ≥ 3}. The cohomology class of this variety of special divisors is computed in [ACGH] p.326:

[C_{g}^{2}] = \frac{θ^{4}}{12} − \frac{xθ^{3}}{3} + \frac{x^{2}θ^{2}}{6} ∈ H^{8}(C_{g−3}, ℚ).

Noting that ι^{∗}(θ) = θ and ι^{∗}(x) = x, one needs to estimate the pull-backs of the tautological monomials x^{α}θ^{4−α}. For this purpose, we use [ACGH] p.358:

χ^{∗}(x^{α}θ^{4−α}) = \frac{g!}{(g − 4 + α)!}[(1 + γt_{1} + t_{2})^{α} · (1 + γ^{2}t_{1} + t_{2})^{4−α}]_{t_{1}t_{2}^{3}},

where the last symbol indicates the coefficient of the monomial t_{1}t_{2}^{3} in the polynomial appearing on the right side of the formula. The rest follows after a routine evaluation.

□

The second enumerative ingredient in the proof of Theorem 0.8 is the following result, which can be proved by degeneration using Schubert calculus:

Proposition 4.2. For a general curve [C] ∈ M_{g−1}, there exist a finite number of pairs (L, x) ∈ W_{g}^{2}(C) × C satisfying the conditions

h^{0}(C, L ⊗ O_{C}(−2x)) ≥ 2, and h^{0}(C, L ⊗ O_{C}(−(g − 2)x)) ≥ 1.

Each pair corresponds to a complete linear series L. The number of such pairs is equal to

n(g − 1) := (g − 1)(g − 2)(g − 3)(g − 4)^{2}.

Proof of Theorem 0.8. We expand [\overline{D}_{g}] ∈ Pic(\overline{M}_{g,g−3}), and begin the calculation by determining the coefficients of λ, δ_{irr} and ∑_{i=1}^{g−3} ψ_{i} respectively. It is useful to observe that if ϕ_{n} : \overline{M}_{g,n} → \overline{M}_{g,n−1} is the map forgetting the marked point labeled by n for some n ≥ 1 and D is any divisor class on \overline{M}_{g,n}, then for distinct labels i, j ≠ n, the λ, δ_{irr} and ψ_{j} coefficients of the divisors D on \overline{M}_{g,n} and (ϕ_{n})_{∗}(D · δ_{0:in}) on \overline{M}_{g,n−1} respectively, coincide. The divisor (ϕ_{n})_{∗}(D · δ_{0:in}) can be thought of as the locus of points [C, x_{1}, …, x_{n}] ∈ D where the points x_{i} and x_{n} are allowed to come together. By iteration, the divisor \overline{D}_{g}^{g−3} on \overline{M}_{g,1} obtained by letting all points x_{1}, …, x_{g−3} coalesce, has the same λ and δ_{irr} coefficients as \overline{D}_{g}. But obviously

\overline{D}_{g}^{g−3} = \{[C, x] ∈ M_{g,1} : ∃L ∈ W_{g}^{2}(C) such that h^{0}(C, L ⊗ O_{C}(−(g − 3)x)) ≥ 1\},

and note that this is a ”pointed Brill-Noether divisor” in the sense of Eisenbud-Harris. The cone of Brill-Noether divisors on \overline{M}_{g,1} is 2-dimensional, see [EH2] Theorem 4.1, and exists constants μ, ν ∈ ℚ, such that \overline{D}_{g}^{g−3} ≡ μ · 𝔅𝔑 + ν · \overline{W}, where

𝔅𝔑 := (g + 3)λ − \frac{g + 1}{6}δ_{irr} − ∑_{j=1}^{g−1} j(g − j)δ_{j:1} ∈ Pic(\overline{M}_{g,1})

is the pull-back from \overline{M}_{g} of the Brill-Noether divisor class and \overline{W} ∈ Pic(\overline{M}_{g,1}) is the class of the Weierstrass divisor. The coefficients μ and ν are computed by intersecting both sides of the previous identity with explicit curves inside \overline{M}_{g,1}. First we fix a genus g curve C and let the marked point vary along C. If C_{x} := ϕ^{−1}([C]) ⊂ \overline{M}_{g,1} denotes the induced curve in moduli, then the only generator of Pic(\overline{M}_{g,1}) which has non-zero intersection number with C_{x} is ψ, and C_{x} · ψ = 2g − 2. On the other hand C_{x} · \overline{D}_{g}^{g−3} = N(g, g − 3), that is, ν = N(g, g − 3)/(g(g − 1)(g + 1)).

To compute μ, we construct a curve inside Δ_{1:1} as follows: Fix a 2-pointed elliptic curve [E, x, y] ∈ M_{1,2} such that the class x − y ∈ Pic^{0}(E) is not torsion, and a general curve [C] ∈ M_{g−1}. We define the family C̄_{1} := {[C ∪_{y} E, x]}_{y∈C}, obtained by varying the point of attachment along C, while keeping the marked point fixed on E. The only generator of Pic(\overline{M}_{g,1}) meeting C̄_{1} non-trivially is δ_{1:1} = δ_{g−1:∅}, in which case C̄_{1} · δ_{1:1} = −2g + 4. On the other hand, C̄_{1} · \overline{D}_{g}^{g−3} is equal to the number of limit linear series 𝔤_{g}^{2} on curves of type C ∪_{y} E, having vanishing sequence at least (0, 1, g − 3) at x ∈ E. This can happen only if this linear series is refined and its C-aspect has vanishing sequence at the point of attachment y ∈ C equal to either (i) (1, 2, g − 3), or (ii) (0, 2, g − 2). In both cases, the E-aspect being uniquely determined, we obtain that C̄_{1} · \overline{D}_{g}^{g−3} = N(g − 1, g − 4) + n(g − 1). This leads to μ = 3(g − 3)(g − 4)/(g + 1).

Next, let \overline{D}_{g}^{g−4} be the divisor on \overline{M}_{g,2} obtained from \overline{D}_{g} by letting all marked

points except one, come together. Precisely, \overline{D}_{g}^{g−4} is the closure of the locus of curves [C, x, y] ∈ M_{g,2} such that there exists L ∈ W_{g}^{2}(C) with h^{0}(C, L⊗O_{C}(−x−(g−4)y)) ≥ 1. We express \overline{D}_{g}^{g−4} ≡ c_{x}ψ_{x}+c_{y}ψ_{y} −eδ_{0:xy} −⋯ ∈ Pic(\overline{M}_{g,2}), and observe that c_{x} equals the ψ-coefficient of \overline{D}_{g}, whereas the coefficient e = ν\binom{g+1}{2} has already been calculated. We fix a general curve [C] ∈ \overline{M}_{g} and define test curves C_{x} := \{[C, x, y] : x ∈ C\} ⊂ \overline{M}_{g,2} and C_{y} := \{[C, x, y] : y ∈ C\} ⊂ \overline{M}_{g,2}, by fixing one general marked point on C and letting the other vary freely. By intersecting \overline{D}_{g}^{g−4} with these curves we obtain the formulas:

(2g−1)c_{x}+c_{y}−e = C_{x}·\overline{D}_{g}^{g−4} = N(g, 1) and c_{x}+(2g−1)c_{y}−e = C_{y}·\overline{D}_{g}^{g−4} = N(g, g−4).

Solving this system, determines c_{x}. Finally, the δ_{0:2}-coefficient of \overline{D}_{g} is computed by intersecting \overline{D}_{g} with the test curve ϕ_{g−3}^{−1}([C, x_{1}, …, x_{g−4}]) ⊂ \overline{M}_{g,g−3}, obtained by fixing g − 4 marked points on a general curve, and letting the remaining point vary. □

As an application, we bound the effective cone of the symmetric product of degree g − 3 on a general curve [C] ∈ M_{g}. As before, let u : C_{g−3} −−→ \overline{C}_{g,g−3} the (rational) fibre map and D̃_{g} the effective divisor on \overline{C}_{g,g−3} to which \overline{D}_{g} descends. Then D_{g}[C] := u^{∗}(D̃_{g}) is an effective divisor on C_{g−3}:

Theorem 4.3. The cohomology class of the codimension one locus inside C_{g−3}

D_{g}[C] := {D ∈ C_{g−3} : ∃L ∈ W_{g}^{2}(C) withh^{0}(C, L ⊗ O_{C}(−D)) ≥ 1} equals

[D_{g}(C)] = \frac{(g − 5)(g − 3)(g − 1)}{3}(θ − \frac{g}{g − 3}x).

It is natural to wonder whether the class θ − \frac{g}{g−3}x is extremal in Eff(C_{g−3}). If so, D_{g}[C] together with the diagonal class δ_{C} ≡ −θ +(2g −4)x would generate the effective cone inside the 2-dimensional space N^{1}(C_{g−3})_{ℚ}. We refer to [K1] Theorem 3, for a proof that δ_{C} spans an extremal ray, which shows that in order to compute Eff(C_{g−3}), one only has to determine the slope of Eff(C_{g−3}) in the fourth quadrant of the (θ, x)-plane. A similar description of the effective cone of C_{g−2} was given in [Mus]. We have a partial result in this direction, showing that all effective divisors of slope higher than \frac{g}{g−3} (if any), must contain a geometric codimension one subvariety of D_{g}[C].

Proposition 4.4. Any irreducible effective divisor on C_{g−3} with class proportional to θ − αx ∈ H^{2}(C_{g−3}, ℚ), where α > \frac{g}{g−3}, contains the codimension two locus inside C_{g−3}

Z_{g−3}[C] := {D ∈ C_{g−3} : ∃A ∈ W_{g−2}^{1}(C) withH^{0}(C, A ⊗ (−D)) ≠ 0}.

Proof. By calculation, note that for A ∈ W_{g−2}^{1}(C), the inequality [V_{g−3}^{1}(A)] · (θ − αx) < 0 holds, whereas [V_{g−3}^{1}(A)] · D_{g}[C] = 0. □

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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