Gavril Farkas

The Kodaira dimension of the moduli space of Prym varieties

Prym varieties provide a correspondence between the moduli spaces of curves and abelian varieties Mg and Ag−1, via the Prym map Pg : Rg → Ag−1 from the moduli space Rg parameterizing pairs [C, η], where [C] ∈ Mg is a smooth curve and η ∈ Pic(C)[2] is a torsion point of order 2. When g ≤ 6 the Prym map is dominant and Rg can be used directly to determine the birational type of Ag−1. It is known that Rg is rational for g = 2, 3, 4 (see [Dol] and references therein and [Ca] for the case of genus 4) and unirational for g = 5 (cf. [IGS] and [V2]). The situation in genus 6 is strikingly beautiful because P6 : R6 → A5 is equidimensional (precisely dim(R6) = dim(A5) = 15). Donagi and Smith showed that P6 is generically finite of degree 27 (cf. [DS]) and the monodromy group equals the Weyl group WE6 describing the incidence correspondence of the 27 lines on a cubic surface (cf. [D1]). There are three different proofs that R6 is unirational (cf. [D1], [MM], [V]). Verra has very recently announced a proof of the unirationality ofR7 (see also Theorem 0.8 for a weaker result). The Prym map Pg is generically injective for g ≥ 7 (cf. [FS]), although never injective. In this range, we may regard Rg as a partial desingularization of the moduli space Pg(Rg) ⊂ Ag−1 of Prym varieties of dimension g − 1.

The Mori cones of moduli spaces of pointed curves of small genus

We compute the Mori cones of the moduli spaces M g,n of n pointed stable curves of genus g, when g and n are relatively small. For instance we show that for g < 14 every curve in Mg is equivalent to an effective combination of the components of the locus of curves with 3g - 4 nodes. We completely describe the cone of nef divisors for the space M 0,6 , thus verifying Fultons conjecture for this space. Using this description we obtain a classification of all the fibrations of M 0,6 .

Green’s conjecture for curves on arbitrary K 3 surfaces

Green’s conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin’s results on syzygies of K 3 sections, to the case of K 3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz–Ramanan, provides a complete solution to Green’s conjecture for smooth curves on arbitrary K 3 surfaces.

Divisors on Mg,g+1 and the minimal resolution conjecture for points on canonical curves

Abstract We use geometrically defined divisors on moduli spaces of pointed curves to compute the graded Betti numbers of general sets of points on any nonhyperelliptic canonically embedded curve. This gives a positive answer to the Minimal Resolution Conjecture in the case of canonical curves. But we prove that the conjecture fails on curves of large degree. These results are related to the existence of theta divisors associated to certain stable vector bundles.

THE MODULI SPACE OF TWISTED CANONICAL DIVISORS

The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in the moduli space of Deligne-Mumford stable pointed curves which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the moduli spaces of canonical divisors. In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus g curves are of pure codimension g in the moduli spaces of stable pointed curves. In addition to the closure of the canonical divisors on nonsingular curves, the moduli spaces have virtual components. In the Appendix, a complete proposal relating the sum of the fundamental classes of all components (with intrinsic multiplicities) to a formula of Pixton is proposed. The result is a precise and explicit conjecture in the tautological ring for the weighted fundamental class of the moduli spaces of twisted canonical divisors. As a consequence of the conjecture, the classes of the closures of the moduli spaces of canonical divisors on nonsingular curves are determined (both in the holomorphic and meromorphic cases).

Gaussian maps, Gieseker-Petri loci and large theta-characteristics

For an integer g ≥ 1 we consider the moduli space Sg of smooth spin curves parametrizing pairs (C,L), where C is a smooth curve of genus g and L is a thetacharacteristic, that is, a line bundle on C such that L2 ∼= KC . It has been known classically that the natural map π : Sg → Mg is finite of degree 2 2g and that Sg is a disjoint union of two components Seven g and S odd g corresponding to even and odd theta-characteristics. A geometrically meaningful compactification Sg of Sg has been constructed by Cornalba by means of stable spin curves of genus g (cf. [C]). The space Sg and more generally the moduli spaces S 1/r g,n of stable n-pointed r-spin curves of genus g, parametrizing pointed curves with r-roots of the canonical bundle, have attracted a lot of attention in recent

Rational maps between moduli spaces of curves and Gieseker-Petri divisors

is injective. The theorem, conjectured by Petri and proved by Gieseker [G] (see [EH3] for a much simplified proof), lies at the cornerstone of the theory of algebraic curves. It implies that the variety Gd(C) = {(L, V ) : L ∈ Pic (C), V ∈ G(r + 1,H0(L))} of linear series of degree d and dimension r is smooth and of expected dimension ρ(g, r, d) := g− (r+1)(g−d+ r) and that the forgetful mapGd(C) → W r d (C) is a rational resolution of singularities (see [ACGH] for many other applications). It is an old open problem to describe the locus GPg ⊂ Mg consisting of curves [C] ∈ Mg such that there exists a line bundle L on C for which the Gieseker-Petri theorem fails. Obviously GPg breaks up into irreducible components depending on the numerical types of linear series. For fixed integers d, r ≥ 1 such that g − d + r ≥ 2, we define the locus GPg,d consisting of curves [C] ∈ Mg such that there exist a pair of linear series (L, V ) ∈ G r d(C) and (KC ⊗ L ∨,W ) ∈ G 2g−2−d (C) for which the multiplication map μ0(V,W ) : V ⊗W → H (C,KC )

The classification of universal Jacobians over the moduli space of curves

We carry out a complete birational classification of the degree g universal Jacobian P_g over the moduli space of curves, highlighting the transition cases g=10, 11. The universal Jacobian is unirational when g 11, the variety P_g has Kodaira dimension 3g-3, that is, the maximum allowed by Iitakas easy addition formula for fibre spaces. In particular, we disprove the expectation that P_g and M_g have the same Kodaira dimension for all genera.

HIGHER RANK BRILL–NOETHER THEORY ON SECTIONS OF K3 SURFACES

We discuss the role of K3 surfaces in the context of Mercats conjecture in higher rank Brill-Noether theory. Using liftings of Koszul classes, we show that Mercats conjecture in rank 2 fails for any number of sections and for any gonality stratum along a Noether-Lefschetz divisor inside the locus of curves lying on K3 surfaces. Then we show that Mercats conjecture in rank 3 fails even for curves lying on K3 surfaces with Picard number 1. Finally, we provide a detailed proof of Mercats conjecture in rank 2 for general curves of genus 11, and describe explicitly the action of the Fourier-Mukai involution on the moduli space M_{11}.

Syzygies of torsion bundles and the geometry of the level l modular variety over M g

We formulate, and in some cases prove, three statements concerning the purity or, more generally the naturality of the resolution of various rings one can attach to a generic curve of genus g and a torsion point of order l in its Jacobian. These statements can be viewed an analogues of Greens Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding non-vanishing locus in the moduli space R_{g,l} of twisted level l curves of genus g and use this to derive results about the birational geometry of R_{g, l}. For instance, we prove that R_{g,3} is a variety of general type when g>11 and the Kodaira dimension of R_{11,3} is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the Prym-Green conjecture in genus 8 and level 2.