THE UNIVERSAL THETA DIVISOR OVER THE MODULI SPACE OF CURVES

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1 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 Ag 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 C g,g 1 := 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], [M] 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 Ng of dimension n g := N g g + and index n g, 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) Vg 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 x g 1 ]. Since Σ is a Grassmann bundle over the rational variety Vg 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 M g,g 1 is uniruled, see [FP], [FV]. For the remaining genera, we prove the following classification result: Theorem 0.. The universal theta divisor Th g is a variety of general type for g 1. We also have a birational classification theorem for the universal degree n symmetric product C g,n := M g,n /S n for all 1 n g, and refer to Section 3 for details. 1

2 G. FARKAS AND A. VERRA Our results are complete in degree g and less precise as n decreases. Similarly to Theorem 0., the nature of C g,g changes when g = 1: Theorem 0.3. The universal degree g symmetric product C g,g is uniruled for g < 1 and a variety of general type for g 1. The proofs of Theorems 0. and 0.3 rely on two ingredients. First, we use our result [FV], stating that for g 4, the singularities of C g,n impose no adjoint conditions, that is, pluricanonical forms defined on the smooth locus of C g,n extend to a smooth model of the symmetric product. Precisely, if ǫ : C g,n C g,n denotes any resolution of singularities, then for any l 0, there is a group isomorphism ǫ : H 0( ) = (C g,n ) reg, K l C H 0 ( Cg,n ), K ec l g,n g,n. In particular, Th g is of general type when the canonical class K Cg,g 1 Pic(C g,g 1 ) is big. This makes the problem of understanding the effective cone of C g,g 1 of some importance. If π : M g,g 1 C g,g 1 is the quotient map, the Hurwitz formula gives that (1) π (K Cg,g 1 ) K Mg,g 1 δ 0: Pic(M g,g 1 ). The sum g 1 ψ i Pic(M g,g 1 ) S g 1 of cotangent tautological classes descends to a big and nef class on C g,g 1 (cf. Proposition 1.), thus in order to conclude that Th g is of general type, it suffices to exhibit an effective divisor D Eff(C g,g 1 ), such that g 1 () π (K Cg,g 1 ) Q >0 λ, ψ i + φ Eff(M g ) + Q 0 π ([D]), δ i:c : i 0, c. In this formula, φ : M g,g 1 M g denotes the morphism forgetting the marked points, and refer to Section 1 for the standard notation for boundary divisor classes on M g,n. Comparing condition () against the formula for K Cg,g 1 given by (4), if one writes π (D) aλ b irr δ irr + c g 1 ψ i i,c b i:cδ i:c Pic(M g,g 1 ), the following inequality (3) 3c < b 0: is a necessary condition for the existence of a divisor D satisfying (). 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 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(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 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 1 g 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 C := {p + D : p C, D C g 3 }, see [An]. In particular, this identification allows one to reconstruct the curve C from the

3 THE UNIVERSAL THETA DIVISOR OVER THE MODULI SPACE OF CURVES 3 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 p) ) 0}. The key ingredient in the proof of Theorem 0. is the calculation of the class of RT g : Theorem 0.4. The closure in M g,g 1 of the locus RT g := {[C, x 1,...,x g 1 ] M g,g 1 : x x g 1 Res C (γ)} is linearly equivalent to, g 1 RT g 4(g 7)λ + 4(g ) ψ i δ irr (1g )δ 0: g i 1 ( ) i 3 5i 3i+4g 4i s+14si 6gs s+s g 3s + δ i:s Pic(M g,g 1 ). i=0 s=0 In particular we note that condition (3) is satisfied. Since by construction, RT g is S g 1 -invariant, it descends to an effective divisor RT g on C g,g 1 which, as it turns out, spans an extremal ray of the cone Eff(C g,g 1 ). Indeed, the universal theta divisor comes equipped with the rational involution τ : C g,g 1 C g,g 1 given by τ ( [C, x x g 1 ] ) := [C, y y g 1 ], where O C (y y g 1 + x x g 1 ) = K C. Then RT g is the pull-back of the boundary divisor 0: C g,g 1 under this map. Since the extremality of 0: is easy to establish, the following result comes naturally: Theorem 0.5. The effective divisor RT g is covered by irreducible curves Γ g C g,g 1 such that Γ g RT g < 0. In particular RT g Eff(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 Wg+1 (C), in particular A is complete and has only ordinary ramification points. The general point of Γ g corresponds to an element [C, D] C g,g 1, where D C g 1 is an effective divisor such that H 0( C, A O C ( p 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 for details. We explain briefly how Theorem 0.4 implies the statement about the Kodaira dimension of C g,g 1. We choose an effective divisor D aλ [g/] i=0 b iδ i Eff(M g ) on the moduli space of curves, with a, b i 0, having slope s(d) := a min i b i as small as possible. Then note that the following linear combination π (K Cg,g 1 ) 1 6g 11 ( 3 [RT g] (1g 5)φ (D) g 1 ψ i ( (84g 185) (1g 5)s(D) ) ) λ is expressible as a positive combination of boundary divisors on M g,g 1. Since, as already pointed out, the class g 1 ψ i Pic(M g,g 1 ) descends to a big class on C g,g 1, one obtains the following:

4 4 G. FARKAS AND A. VERRA Corollary 0.6. For all g such that the slope of the moduli space of curves satisfies the inequality 84g 185 s(m g ) := inf D Eff(Mg) s(d) < 1g 5, the universal theta divisor Th g is of general type. The bound appearing in Corollary 0.6 holds precisely when g 1; for g such that g + 1 is composite, the inequality s(m g ) 6 + 1/(g + 1) is well-known, and D can be chosen to be a Brill-Noether divisor M r g,d corresponding to curves with a g r d when the Brill-Noether number ρ(g, r, d) = 1, cf. [EH1]. When g + 1 is prime and g 1, then in practice g = k 16, and D can be chosen to be the Gieseker-Petri GP 1 g,k consisting of curves C possessing a pencil A Wk 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 = 1, one has to use the divisor constructed on M 1 in [FV1]. Finally, when g 11 it is known that s(m g ) 6 + 1/(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 -invariant extremal ray of the cone of effective divisors on M g,g. 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 M g,n with 1 n g, cf. Section 3. Theorem 0.7. The closure inside M g,g of the locus F g,1 := {[C, x 1,...,x g ] M g,g : A Wg 1(C) 1 with H 0( g C, A( x i ) ) 0} is a non-movable, extremal effective divisor on M g,g. Its class is given by the formula: g F g,1 (g 1)λ+(g 3) ψ i δ irr 1 g s(g 4+sg s) δ 0:s Pic(M g,g ). s= Note that again, inequality (3) is satisfied, hence F g,1 can be used to prove that K Cg,g is big. Moreover, F g,1 descends to an extremal divisor F g,1 Eff(C g,g ). In fact, we shall show that F 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 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) Wg+1 (C). A natural codimension one condition on 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 M g,g 3 of the locus D g := {[C, x 1,...,x g 3 ] M g,g 3 : L Wg (C) with H 0( g 3 C, L( x i ) ) 0}

5 THE UNIVERSAL THETA DIVISOR OVER THE MODULI SPACE OF CURVES 5 is an effective divisor. Its class in Pic(M g,g 3 ) is equal to ( ) (g 17) g 3 D g λ+ g 3 ( ) g 4 g 3 ( ) g 3 ψ i δ irr (g 5g+5)(g 5)δ 0: CONES OF DIVISORS ON UNIVERSAL SYMMETRIC PRODUCTS The aim of this section is to establish certain facts about boundary divisors on M g,n and C g,n, see [AC] for a standard reference. We follow the convention set in [FV], that is, if M is a Deligne-Mumford stack, we denote by M its coarse moduli space. For an integer 0 i [g/] and a subset T {1,...,n}, we denote by i:t the closure in M g,n of the locus of n-pointed curves [C 1 C, x 1,...,x n ], where C 1 and C 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 := [ i:t ] Q Pic(M g,n ). For 0 i [g/] and 0 s g, we set i:s := δ i:t, δ i:s := [ i:s ] Q Pic(M g,n ). #(T)=s By convention, δ 0:s :=, for s <, and δ i:s := δ g i:n s. If φ : M g,n M g is the morphism forgetting the marked points, we set λ := φ (λ) and δ irr := φ (δ irr ), where δ irr := [ irr ] Pic(M g ) denotes the class of the locus of irreducible nodal curves. Furthermore, ψ 1,...,ψ n Pic(M g,n ) are the cotangent classes corresponding to the marked points. The canonical class of M g,n is computed via Kodaira-Spencer theory: n (4) K Mg,n 13λ δ irr + ψ i δ i:t δ 1: Pic(M g,n ). T {1,...,n} i 0 Let C g,n := M g,n /S n be the universal symmetric product and π : M g,n C g,n (respectively ϕ : C g,n M g ) the projection (respectively the forgetful map), so that φ = ϕ π. We denote by λ, δ irr, δ i:c := [ i:c ] Pic(C g,n ) the divisor classes on the symmetric product pulling-back to the same symbols on M g,n. Clearly, π ( λ) = λ, π ( δ irr ) = δ irr, π ( δ i:c ) = δ i:c ; in the case i = 0, c =, this reflects the branching of the map π along the divisor 0: C g,n. Following [FV], let L denote the line bundle on C g,n, having fibre L[C, x x n ] := T x 1 (C) T x n (C), over a point [C, x x n ] := π([c, x 1,...,x n ]) C g,n. We set ψ := c 1 (L), and note: (5) π ( ψ) n ( ) n n = ψ i δ 0:T = ψ i s δ 0:s Pic(M g,n ). i T {1,...,n} Proposition 1.1. For g 3 and n 0, the morphism π : Pic(C g,n ) Q Pic(M g,n ) Q is injective. Furthermore, there is an isomorphism of groups Pic(C g,n ) Q = N 1 (C g,n ) Q. Proof. The first assertion is an immediate consequence of the existence of the norm morphism Nm π : Pic(M g,n ) Pic(C g,n ), such that Nm π (π (L)) = L deg(π), for every L Pic(C g,n ). The second part comes from the isomorphism Pic(M g,n ) Q = N 1 (M g,n ) Q, s=

6 6 G. FARKAS AND A. VERRA coupled with the commutativity of the obvious diagrams relating the Picard and Néron- Severi groups of M g,n and C g,n respectively. One may thus identify Pic(C g,n ) Q = Pic(Mg,n ) Sn Q. The Riemann-Hurwitz formula applied to the branched covering π : M g,n C g,n yields, n n π (K ) = K Cg,n M g,n δ 0: 13λ + ψ i δ irr 3δ 0: δ 0:s. As expected, the sum of cotangent classes descends to a big line bundle on C g,n. Proposition 1.. The divisor class N g,n := ψ + n s= s δ 0:s Eff(C g,n ) is big and nef. Proof. The class N g,n is characterized by the property that π (N g,n ) = n ψ i. This is a nef class on M g,n, in particular, N g,n is nef on C g,n. To establish that N g,n is big, we express it as a combination of effective classes and the class κ 1 Pic(C g,n ), where π ( κ 1 ) = κ 1 = 1λ + n ψ i δ irr [g/] i=0 s 0 s=3 δ i:s Pic(M g,n ). Since π ( κ 1 ) is ample on M g,n, it follows that κ 1 is ample as well. To finish the proof, we exhibit a suitable effective class on 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 : M g,n M g,1 the morphism forgetting all but the i-th point, and let ( ) g + 1 g 1 ( ) g i + 1 W λ + ψ δ i:1 Eff(M g,1 ), be the class of the divisor of Weierstrass points on the universal curve. Then one finds n ( ) n ( ) n g + 1 g + 1 π (W g,n ) σi (W) = nλ + ψ i sδ 0:s Pic(M g,n ), and W g,n g λ + ( ) g+1 [g/] ψ s 0 b i:s δ 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 κ 1, the effective class [W g,n ] and other boundary divisor classes. In particular, N g,n is big.. THE UNIVERSAL RAMIFICATION LOCUS OF THE GAUSS MAP We begin the calculation of the divisor 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 (1) g,g 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 (1) g,g 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 ip M (1) g,g 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 (1) g,g 1 M g,1 (respectively s=

7 THE UNIVERSAL THETA DIVISOR OVER THE MODULI SPACE OF CURVES 7 σ p : M (1) g,g 1 M g,1), obtained by forgetting all marked points except x i (respectively p), and then set K i := σi (ω φ) Pic(M (1) g,g 1 ) and K p := σp(ω φ ) Pic(M (1) g,g 1 ). We consider the following cartesian diagram of stacks X f M g,1 q M (1) φ g,g 1 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 (1) g,g 1 X as well as r p : M (1) g,g 1 X, and set E i := Im(r i ), E p := Im(r p ). Thus {E i } g 1 and E p are relative divisors over q. For a point ( [C, x 1,...,x g 1 ], p ) M (1) g,g 1, we denote D := g 1 x i + p C g+1, and have the following exact sequence: 0 H0( O C (D) ) H 0 H 0( O D (D) ) α D H 1 (O C ) H 1( O C (D) ) 0. (O C ) In particular, the morphisms α D globalize to a morphism of vector bundles over M (1) g,g 1 M g ( ( g 1 ) ) α : A := q O X E i + E p /OX R 1 q O X. The subvariety Z := {( [C, x 1,...,x g 1 ], p ) M (1) g,g 1 : H0( K C ( p 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 ( A ( R 1 q O X ) ) g 1 = c ( q! O X ( ) E i + E p ) A (M (1) g,g 1 ), where the last term can be computed by Grothendieck-Riemann-Roch: ( ch (q g 1 ) ) [( ( g 1! O X E i + E p = q E i + E p ) k ) ( 1 c 1(ω q ) + c 1 (ω )] q) +, k! 1 k 0 and we are interested in evaluating the terms of degree 1 and in this expression. The result of applying GRR to the morphism q, can be summarized as follows: Lemma.1. One has the following relations in A (M (1) g,g 1 ): (i) ( g 1 ch 1 (q OX ( E i + E p ) )) g 1 g 1 = λ K i 3K p + ip. (ii) ( g 1 ch (q OX ( E i + E p ) )) = 5 K p + 1 g 1 Ki (K i + K p ) ip. g 1

8 8 G. FARKAS AND A. VERRA Proof. We apply systematically the push-pull formula and the following identities: E i = E i q (K i ), E p = 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 ( ip ), and q (c 1(ω q )) = 1λ. Proposition.. The formula RT g 4(g 7)λ + (4g 8) g 1 ψ i Pic(M g,g 1 ) holds. Proof. We apply the results of Lemma.1, as well as the formulas from [HM] p. 55, in order to estimate the push-forward under u of the degree monomials in tautological classes. Setting F := q ( OX ( g 1 E i + E p ) ), we obtain that ( u ch 1 (F) ) g 1 ( = (8g 116)λ + (8g 4) ψ i, and u ch (F) ) g 1 = 30λ 4 ψ i, hence [RT g ] = u ( ch 1 (F) ch (F) ) /, and the claimed formula follows at once. We proceed now towards proving Theorem 0.4 and expand the divisor class [RT g ] Pic(M g,g 1 ) in the standard basis of the Picard group, that is, g 1 g i 1 RT g aλ + c ψ i b irr δ irr b i:s δ i:s. i=0 s=0 We have just computed a = 4(g 7) and c = 4(g ). The remaining coefficients are determined by intersecting RT g with curves lying in the boundary of M g,g 1 and understanding how RT g degenerates. We begin with the coefficient b 0: : Proposition.3. One has the relation (4g 6)c (g )b 0: = (4g )(g ). It follows that b 0: = 1g. Proof. We fix a general pointed curve [C, x 1,...,x g ] M g,g and consider the family C xg 1 := { [C, x 1,...,x g, x g 1 ] : x g 1 C } M g,g 1. The curve C xg 1 is the fibre over [C, x 1,...,x g ] of the morphism M g,g 1 M g,g forgetting the point labeled by x g 1. Note that C xg 1 ψ i = 1 for i = 1,...,g and C xg 1 ψ g 1 = 3g 4 = g + (g ). Obviously C xi δ 0: = g 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. The intersection of C xi with the remaining generators of Pic(M g,g 1 ) is equal to zero. We set A := K C O C ( x 1 x g ) Wg 1 (C). By the generality assumption, h 0 (C, A) =, and all ramification points of A are simple. Pointed curves in the intersection C xg 1 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 ( p x g 1 ) ) 0. The pencil A carries 4g ramification points. For each of them there are g 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 1b irr + b 1:0 = 0 holds. Indeed, the divisor RT g is disjoint from the curve in 1:0 M g,g 1, obtained from a fixed pointed curve [C, x 1,...,x g 1, q] 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.

9 THE UNIVERSAL THETA DIVISOR OVER THE MODULI SPACE OF CURVES 9 Proposition.4. One has the relation b irr =. Proof. We fix a general curve [C, q, x 1,...,x g 1 ] M g 1,g, and we define the family C irr := { [C/t q, x 1,...,x g 1 ] : t C} irr M g,g 1. Then C irr ψ i = 1 for i = 1,...,g 1, C irr δ irr = (deg(k C ) + ) = g +, and finally C irr δ 1:0 = 1. All other intersection numbers with generators of Pic(M g,g 1 ) equal zero. We fix an effective divisor D C e of degree e g (for instance D = q + g 1 x i). For each pair of points (t, p) C C, there is an exact sequence on C g 1 0 H 0( C, K C (q + t p x i ) ) H 0( C, K C (D + q + t p x i ) ) β t,p g 1 H 0( D, K C (D + q + t p x i ) ) H 1( C, K C (q + t p x i ) ) 0. The intersection C irr 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 g 1 x i) Pic g (C). We denote by 1, 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 ( 1 13 ). Applying the Porteous formula, one can write C irr RT g = c (R 1 f F R 0 f F) = ch 1(f! F) + ch (f! F) A (C C). We evaluate ch i (f! F) using GRR applied to the morphism f, that is, ( ) [( p a ch(f! F) = f 1 (A) + 1 ) ( )] a! p 1(K C ). a 0 Denoting by F 1, F H (C C) the class of the fibres, after calculations one finds that g 1 ch 1 (f F) = (g )F 1 4(g )F C H (C C, Q), ch (f F) = (g ) H 4 (C C, Q), that is, c (R 1 f F R 0 f F) = 4(g )(g 1). Coupled with (6), this yields b irr =. We are left with the task of determining the coefficient of δ i:s in the expansion of [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 ] M i,s and [D, q, x s+1,...,x g 1 ] 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 } i:s M g,g 1. We summarize the non-zero intersection numbers of C i:s with generators of Pic(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 = i + s. g 1

10 10 G. FARKAS AND A. VERRA Theorem.5. We fix integers 0 i g and 0 s i 1. Then, the following formula holds: b i:s = i 3 5i 3i + 4g 4i s + 14si 6gs s + s g 3s +. In the proof an essential role is played by the following calculation: Proposition.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 p) ) 0, is equal to a(i, s) := (i s 1)(i 3 5i + i + i s + 3is). Remark.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) = 34 equals the number of canonical divisors of type 3q + p + x K C, where [C] M 4. This matches de Jonquières formula, cf. [ACGH] p.359. Proof of Theorem.5. We fix a general point [C q D, x 1,...,x g 1 ] C i:s 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 (iq) ) and ω C H 0( C, K C O C (g i)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 x g 1. The condition ord q (ω C ) + ord q (ω D ) g, 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 x s, div(ω D ) x s x g 1 + p. Since the points q, x s+1,...,x g 1 D are general, we find that ord q (ω D ) i + s. Moreover, K D O D ((i s+)q x s+1 x g 1 ) Wg 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 ) g 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 ( s j=1 x j) Wi 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 1 is). Multiplying this with the number of choices for p D, we obtain a total contribution of 4(g i)(i s)(i is 1) to the intersection C i:s 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 [EH] 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 ) = g i s 1, therefore 0 σ C H 0( s C, K C O C ( x j (i s 1)q p) ). 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. j=1

11 THE UNIVERSAL THETA DIVISOR OVER THE MODULI SPACE OF CURVES 11 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 q1 E q D, obtained from C D by inserting a smooth rational component E at q, where {q 1 } := C E and {q } := D E. There also exist non-zero sections ω D H 0( D, K D (iq ) ), ω E H 0( E, O E (g ) ), ω C H 0( C, K C ((g i)q 1 ) ), satisfying ord q1 (ω C ) + ord q1 (ω E ) g and ord q (ω E ) + ord q (ω D ) g. Furthermore, ω E vanishes doubly at a point p {q 1, q } c. Since ω C (respectively ω D ) also vanishes along the divisor x x s (respectively x s x g 1 ), it follows that ord q1 (ω C ) g i s and ord q (ω D ) i + s 1, hence by compatibility, ord q1 (ω E ) + ord q (ω E ) g 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 [RT g ] is equal to 4g 8, we find the relation (7) (i + s)b i:s sb i:s 1 + s(4g 8) = 4(g i)(s 1)(si i + ) + a(i, s). For s = 0, we have by convention b i: 1 = 0, which gives b i:0 = i 3 5i 3i + 4g + 1. By induction, we find using recursion (7) the claimed formula for b i:s. As already explained, having calculated the class [RT g ] Pic(M g,g 1 ) and using known bound on the slope s(m g ), one derives that Th g is of general type when g 1. 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 Vg g 1. When g {10, 11}, the space M g,g 1 is uniruled [FP]. This implies the uniruledness of Th g as well. 3. THE KODAIRA DIMENSION OF C g,n In this section we provide results concerning the Kodaira dimension of the symmetric product C g,n, where n g. 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)/ on a curve [C] M g. We set integers g 1 and 1 m g/, then consider the locus F g,m := {[C, x 1,...,x g m ] M g,g m : A Wg m(c) 1 with H 0( g m C, A( x j ) ) 0}. A parameter count shows that F g,m is expected to be an effective divisor on M g,g m. We shall prove this, then compute the class of its closure in M g,g m. Theorem 3.1. Fix integers g 1 and 1 m g/, then set n := g m and d := g m. The class of the compactification inside M g,g m of the divisor F g,m is given by the formula: ( ( ) 10n g F g,m n ( ) g ) λ + n 1 ( ) n g 1 ψ j n ( ) g δ irr g d 1 g d g 1 d 1 g d 1 j=1 j=1

12 1 G. FARKAS AND A. VERRA n s= s(n g + sgn sn) (g 1)(g d) ( g 1 d ) δ 0:s Pic(M g,n ). Proof. We fix a general curve [C] M g and consider the incidence correspondence Σ := {(D, A) C g m W 1 g m(c) : H 0 (C, A O C ( D)) 0}, together with the projection π 1 : Σ C g m. It follows from [F1] Theorem 0.5, that Σ is pure of dimension g m 1 ( = ρ(g, 1, g m) + 1 ). To conclude that F g,m is a divisor inside M g,g m, it suffices to show that the general fibre of the map π 1 is finite, which implies that φ 1 ([C]) 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 m) p, where p C. One needs to show that for a general curve [C] M g, there exist finitely many pencils A W 1 g m(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 m)-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(, g m+1). In particular their number is finite. In order to compute the class [F g,m ], we expand it in the usual basis of Pic(M g,n ) g m F g,m aλ + c 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 [F] Theorem 4.9. The coefficient b 0: is determined by intersecting F g,m with a fibral curve C xn := {[C, x 1,...,x n 1, x n ] : x n C} M g,n, corresponding to a general (n 1)-pointed curve [C, x 1,...,x n 1 ] 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 xn 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 m 1) ( g m), see [HM] Theorem A, that is, (g + n 4)c (n 1)b 0: = C xn F g,m = { (m + 1) # A Wg m(c) 1 : h 0( C, A O C ( (g m 1)q) ) } ( ) g 1 = (g m 1), m which determines b 0:. The coefficients b 0:s are computed recursively, by exhibiting an explicit test curve Γ 0:s 0:s which is disjoint from F g,m. We fix a general element [C, q, x s+1,...,x n ] M g,n+1 s and a general s-pointed rational curve [P 1, x 1,...,x s ] 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 } 0:s M g,n. Clearly, Γ 0:s F g,m = s c (s ) b 0:s + s b 0:s 1. We claim Γ 0:s F g,m =. Assume that on the contrary, one can find a point q P 1 and a limit linear series g 1 d on P1 q C, l = ( (A, V C ), (O P 1(d), V P 1) ) G 1 d (C) G1 d (P1 ),

13 THE UNIVERSAL THETA DIVISOR OVER THE MODULI SPACE OF CURVES 13 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 x n, div(σ P 1) x 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 1 g m(c). Indeed, by letting all points x s+1,...,x n, q C coalesce, the statement H 0( C, A O C ( (g m) q) ) = 0, for a general [C, q] 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 F g,m = sc + (s )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/] 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 m. 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 m. We denote by θ H (C g m, Q) the class of the pull-back of the theta divisor, and by x H (C g m, Q) the class of the locus {p 0 + D : D C g m 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 m ) Q is generated by x and θ, see [ACGH]. Let Fg,m be the effective divisor on C g,g m to which F g,m descends, that is, π ( F g,m ) = F g,m. The class of F g,m is completely determined by Theorem 3.1. Corollary 3.. Let [C] M g be a general curve. The cohomology class of the divisor F g,m [C] := {D C g m : A Wg m(c) 1 such that H 0 (C, A O C ( D)) 0} is equal to (1 m g )( )( g m θ g g m x). In particular, the class θ g g m x N1 (C g m ) Q is effective. Proof. Let u : C g m C g,g m be the rational map given by u(x x g m ) = [C, x x g m ]. Note that u is well-defined outside the codimension locus of effective divisors with support of length at most g m. We have that u ( δ 0: ) = δ C, where δ C := [ C ]/ is the reduced diagonal. Its class is given by the MacDonald formula, cf. [K1] Lemma 7: δ C θ + (g + d 1)x θ + (g m 1)x. Furthermore, u ( ψ) θ + δ C + (g n 1)x, see [K] Proposition.7. Thus F g,m [C] u ([ F g,m ]), and the conclusion follows after some calculations. The divisor F 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 1 g m(c) with only simple ramification points. The variety of secant divisors V 1 g m(a) := {D C g m : H 0 (C, A O C ( D)) 0}

14 14 G. FARKAS AND A. VERRA is a curve (see [F1]), disjoint from the indeterminacy locus of the rational map u : C g m C g,g m. We set Γ g m (A) := u(v 1 g m (A)) C g,g m. By varying [C] M g and A W 1 g m(c), the curves Γ g m (A) fill-up the divisor F g,m. It is natural to test the extremality of F g,m by computing the intersection number Γ g m (A) F g,m. To state the next result in a unified form, we adopt the convention ( a b) := 0, whenever b < 0. Proposition 3.3. For all integers 1 m < g/, we have the formula: ( )( ) Γ g m (A) F g m g g,m = (m 1). m m In particular, Γ g (A) F g,1 = 0, and the divisor F g,1 Eff(C g,g ) is extremal. Proof. This is an immediate application of Corollary 3.. The class [Vg m 1 (A)] can be computed using Porteous formula, see [ACGH] p.34: g m 1 ( ) m 1 [Vg m(a)] 1 xj θ g m j 1 j (g m 1 j)! H(g m 1) (C g m, Q). j=0 Using the push-pull formula, we write Γ g m (A) F g,m = F g,m [C] [Vg m 1 (A)], then estimate the product using the identity x k θ g m k = g!/(m+k)! H (g m) (C g m, Q) for 0 k g m. For m = 1, observe that Γ g (A) F g,1 = 0. Since the curves of type Γ g (A) cover F g,1, this implies that [ F g,1 ] Eff(C g,g ) generates an extremal ray. We can use Theorem 3.1 to describe the birational type of C g,n when 1 g 1 and 1 n g. We recall that when g 9, the space 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 C g,g, are discussed in detail in [FV]. Furthermore 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, C g,n is of general type in all the cases when M g,n is known to be of general type, see [Log], [F]. We note however that the divisors F g,m only carry one a certain distance towards a full solution. The classification of C g,n is complete only when n {g 1, g, g}. Theorem 3.4. For integers g = 1,...,1, the universal symmetric product C g,n is of general type for all f(g) n g 1, where f(g) is described in the following table. g f(g) Proof. The strategy described in the Introduction to prove that K Cg,g 1 is big, applies to the other spaces C g,n, with 1 n g as well. To show that C g,n is of general type, it suffices to produce an effective class on C g,n which pulls back via π to aλ + c n ψ i b irr δ irr i,s b i:sδ i:s Eff(M g,n ) Sn, such that the following conditions are fulfilled: a + s(m g ) ( ) c b irr (8) < 1 and b 0: 13c 3c > 1. When g n is even, we write g n = m, and for all entries in the table above one can express K Cg,n as a positive combination of n ψ i, [F g,m ], ϕ (D), where D Eff(M g ), and other boundary classes.

15 THE UNIVERSAL THETA DIVISOR OVER THE MODULI SPACE OF CURVES 15 If g n = m + 1 with m Z 0, for each integer 1 j n + 1, we denote by φ j : M g,n+1 M g,n the projection forgetting the j-th marked point and consider the effective S n -invariant effective Q-divisor on M g,n E := 1 n+1 ( ) (φ j ) Fg,m δ n + 1 0:{j,n+1} Eff(Mg,n ). j=1 Using Theorem 3.1 as well as elementary properties of push-forwards of tautological classes, K Cg,n is expressible as a positive Q-combination of boundaries, [E], a pull-back of an effective divisor on M g, and the big and nef class n ψ i precisely in the cases appearing in the table. Remark 3.5. When g / {1, 16, 18}, the bound s(m g ) 6 + 1/(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(m 1 ) = 4415/64 < 6 + 1/13 (see [FV1]), and s(m 16 ) = 407/61 < 6 + 1/17 see [F], coming from Koszul divisors on M 1 and M 16 respectively. On M 18, we use the estimate s(m 18 ) 30/45 given by the class of the Petri divisor GP 1 18,10, see [EH1]. Improvements on the estimate on s(m g ) in the other cases, will naturally translate in improvements in the statement of Theorem AN EFFECTIVE DIVISOR ON 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 [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 (C) C such that Their number is computed by the formula H 0( C, L O C ( γ x (g 3 γ) p) ) 1. N(g, γ) := g(g 1)(g 5) γ(γg 3γ 1). 3 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] ), where C g := {D C g : dim D 3}. The cohomology class of this variety of special divisors is computed in [ACGH] p.36: [Cg] = θ4 1 xθ3 3 + x θ H 8 (C g 3, Q). 6 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 g! [ (1 ) α ( ) = + γt1 + t 1 + γ ) 4 α ] t 1 + t (g 4 + α)! t 1 t, 3 where the last symbol indicates the coefficient of the monomial t 1 t 3 in the polynomial appearing on the right side of the formula. The rest follows after a routine evaluation.

16 16 G. FARKAS AND A. VERRA 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.. For a general curve [C] M g 1, there exist a finite number of pairs (L, x) W g (C) C satisfying the conditions h 0( C, L O C ( x) ), and h 0( C, L O C ( (g )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 )(g 3)(g 4). Proof of Theorem 0.8. We expand [D g ] Pic(M g,g 3 ), and begin the calculation by determining the coefficients of λ, δ irr and g 3 ψ i respectively. It is useful to observe that if φ n : M g,n 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 M g,n, then for distinct labels i, j n, the λ, δ irr and ψ j coefficients of the divisors D on M g,n and (φ n ) (D δ 0:in ) on 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 D g 3 g on M g,1 obtained by letting all points x 1,...,x g 3 coalesce, has the same λ and δ irr coefficients as D g. But obviously D g 3 g = {[C, x] M g,1 : L Wg (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 M g,1 is -dimensional, see [EH] Theorem 4.1, and exists constants µ, ν Q, such that D g 3 g µ BN + ν W, where BN := (g + 3)λ g δ g 1 irr j(g j)δ j:1 Pic(M g,1 ) is the pull-back from M g of the Brill-Noether divisor class and W Pic(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 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]) M g,1 denotes the induced curve in moduli, then the only generator of Pic(M g,1 ) which has non-zero intersection number with C x is ψ, and C x ψ = g. On the other hand C x D g 3 g = N(g, g 3), that is, ν = N(g, g 3)/(g(g 1)(g + 1)). To compute µ, we construct a curve inside 1:1 as follows: Fix a -pointed elliptic curve [E, x, y] M 1, such that the class x y Pic 0 (E) is not torsion, and a general curve [C] M g 1. We define the family C 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(M g,1 ) meeting C 1 non-trivially is δ 1:1 = δ g 1:, in which case C 1 δ 1:1 = g + 4. On the other hand, C1 D g 3 g is equal to the number of limit linear series g g 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,, g 3), or (ii) (0,, g ). In both cases, the E-aspect being uniquely determined, we obtain that C 1 D g 3 g = N(g 1, g 4) + n(g 1). This leads to µ = 3(g 3)(g 4)/(g + 1). j=1

17 THE UNIVERSAL THETA DIVISOR OVER THE MODULI SPACE OF CURVES 17 Next, let D g 4 g be the divisor on M g, obtained from D g by letting all marked points except one, come together. Precisely, D g 4 g is the closure of the locus of curves [C, x, y] M g, such that there exists L Wg (C) with h 0( C, L O C ( x (g 4)y) ) 1. c x ψ x +c y ψ y eδ 0:xy Pic(M g, ), and observe that c x equals the ψ-coefficient of D g, whereas the coefficient e = ν ( ) g+1 has already been calculated. We fix a general curve [C] M g and define test curves C x := {[C, x, y] : x C} M g, and C y := {[C, x, y] : y C} M g,, by fixing one general marked point on C and letting We express D g 4 g the other vary freely. By intersecting D g 4 g (g 1)c x +c y e = C x D g 4 g with these curves we obtain the formulas: = N(g, 1) and c x +(g 1)c y e = C y D g 4 g = N(g, g 4). Solving this system, determines c x. Finally, the δ 0: -coefficient of D g is computed by intersecting D g with the test curve φ 1 g 3 ([C, x 1,...,x g 4 ]) 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 C g,g 3 the (rational) fibre map and D g the effective divisor on C g,g 3 to which D g descends. Then D g [C] := u ( D 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 Wg (C) with h 0( C, L O C ( D) ) 1} (g 5)(g 3)(g 1) ( [D g (C)] = θ g ) 3 g 3 x. equals It is natural to wonder whether the class θ g g 3 x is extremal in Eff(C g 3). If so, D g [C] together with the diagonal class δ C θ+(g 4)x would generate the effective cone inside the -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 was given in [Mus]. We have a partial result in this direction, showing that all effective divisors of slope higher than any), must contain a geometric codimension one subvariety of D g [C]. g g 3 (if Proposition 4.4. Any irreducible effective divisor on C g 3 with class proportional to θ αx H (C g 3, Q), where α > g g 3, contains the codimension two locus inside C g 3 Z g 3 [C] := {D C g 3 : A W 1 g (C) with H 0 (C, A ( D)) 0}. Proof. By calculation, note that for A Wg 1 (C), the inequality [V 1 holds, whereas [Vg 3 1 (A)] D g[c] = 0. REFERENCES g 3 (A)] (θ αx) < 0 [AC] E. Arbarello and M. Cornalba, The Picard groups of the moduli spaces of curves, Topology 6 (1987), [ACGH] E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of algebraic curves, Grundlehren der mathematischen Wissenschaften 67, Springer. [An] A. Andreotti, On a theorem of Torelli, American Journal of Math. 80 (1958),

18 18 G. FARKAS AND A. VERRA [EH1] D. Eisenbud and J. Harris, The Kodaira dimension of the moduli space of curves of genus 3 Inventiones Math. 90 (1987), [EH] D. Eisenbud and J. Harris, Irreducibility of some families of linear series with Brill-Noether number 1, Annales Scientifique École Normale Supérieure (1989), [F1] G. Farkas, Higher ramification and varieties of secant divisors on the generic curve, Journal of the London Mathematical Society 78 (008), [F] G. Farkas, Koszul divisors on moduli spaces of curves, American Journal of Math. 131 (009), [FP] G. Farkas and M. Popa, Effective divisors on M g, curves on K3 surfaces and the Slope Conjecture, Journal of Algebraic Geometry 14 (005), [FV1] G. Farkas and A. Verra, The geometry of the moduli space of odd spin curves, arxiv: [FV] G. Farkas and A. Verra, The classification of the universal Jacobian over the moduli space of curves, arxiv: [FV3] G. Farkas and A. Verra, The universal theta divisor over A 5, in preparation. [Fr] E. Freitag, Die Kodairadimension von Körpern automorpher Funktionen, J. reine angew. Mathematik 96 (1977), [HM] J. Harris and D. Mumford, On the Kodaira dimension of M g, Inventiones Math. 67 (198), [K1] A. Kouvidakis, Divisors on symmetric products of curves, Transactions American Math. Soc. 337 (1993), [K] A. Kouvidakis, On some results of Morita and their application to questions of ampleness, Math. Zeitschrift 41 (00), [Log] A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, American Journal of [M1] Math. 15 (003), S. Mukai, Fano 3-folds, in: Complex Projective Geometry (Trieste/Bergen 1989), London Math. Society Lecture Notes Series, 179, [M] S. Mukai, Curves and symmetric spaces I, American Journal of Math. 117 (1995), [Mum] D. Mumford, On the Kodaira dimension of the Siegel modular variety, in: Algebraic Geometry-Open problems (Ravello 198), Lecture Notes in Mathematics 997 (1983), [Mus] Y. Mustopa, Kernel bundles, syzygies and the effective cone of C g, International Math. Research Notices 010, doi: /imrn/rnq119. [T] Y.S. Tai, On the Kodaira dimension of the moduli space of abelian varieties, Inventiones Math. 69 (198), [Ve] [Vi] A. Verra, On the universal principally polarized abelian variety of dimension 4, in: Curves and abelian varieties (Athens, Georgia, 007) Contemporary Mathematics 465 (008), E. Viehweg, Die Additivität der Kodaira Dimension für projektive Fasserräume über Varietäten des allgemeinen Typs, J. reine angew. Mathematik, 330 (198), HUMBOLDT-UNIVERSITÄT ZU BERLIN, INSTITUT FÜR MATHEMATIK, UNTER DEN LINDEN BERLIN, GERMANY address: farkas@math.hu-berlin.de UNIVERSITÁ ROMA TRE, DIPARTIMENTO DI MATEMATICA, LARGO SAN LEONARDO MURIALDO ROMA, ITALY address: verra@mat.unirom3.it