Giuseppe Pareschi

Regularity on abelian varieties I

We introduce the notion of Mukai regularity (

GV-SHEAVES, FOURIER-MUKAI TRANSFORM, AND GENERIC VANISHING

M

Castelnuovo theory and the geometric Schottky problem

-regularity) for coherent sheaves on abelian varieties. The definition is based on the Fourier-Mukai transform, and in a special case depending on the choice of a polarization it parallels and strengthens the usual Castelnuovo-Mumford regularity. Mukai regularity has a large number of applications, ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties, to higher dimensional type statements and to a study of special classes of vector bundles. Some of these applications are explained here, while others are the subject of upcoming sequels.

Regularity on abelian varieties II: Basic results on linear series and defining equations

We prove a formal criterion for generic vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon, but in the context of an arbitrary Fourier-Mukai correspondence. For smooth projective varieties we apply this to deduce a Kodaira-type generic vanishing theorem for adjoint bundles associated to nef line bundles, and in fact a more general generic Nadel-type vanishing theorem for multiplier ideal sheaves. Still in the context of the Picard variety, the same method gives various other generic vanishing results, by reduction to standard vanishing theorems. We further use our criterion in order to address some examples related to generic vanishing on higher rank moduli spaces.

On the bicanonical map of irregular varieties

We prove and conjecture results which show that Castelnuovo theory in projective space has a close analogue for abelian varieties. This is related to the geometric Schottky problem: our main result is that a principally polarized abelian variety satisfies a precise version of the Castelnuovo Lemma if and only if it is a Jacobian. This result has a surprising connection to the Trisecant Conjecture. We also give a genus bound for curves in abelian varieties.

Components of the Hilbert scheme of smooth space curves with the expected number of moduli

We apply the theory of M-regularity developed by the authors [Regularity on abelian varieties, I, J. Amer. Math. Soc. 16 (2003), 285-302] to the study of linear series given by multiples of ample line bundles on abelian varieties. We define an invariant of a line bundle, called M-regularity index, which governs the higher order properties and (partly conjecturally) the defining equations of such embeddings. We prove a general result on the behavior of the defining equations and higher syzygies in embeddings given by multiples of ample bundles whose base locus has no fixed components, extending a conjecture of Lazarsfeld [proved in Syzygies of abelian varieties, J. Amer. Math. Soc. 13 (2000), 651-664]. This approach also unifies essentially all the previously known results in this area, and is based on Fourier-Mukai techniques rather than representations of theta groups.

Some bounds for the regular genus of PL-manifolds

From the point of view of uniform bounds for the birationality of pluri- canonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this pa- per we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional gen- eralization to this context of curves of genus 2: varieties birationally equivalent to the theta-divisor of an indecomposable principally polar- ized abelian variety. The proof is based on the (generalized) Fourier- Mukai transform.

Hyperplane sections of abelian surfaces

For any d and g such that the Brill-Noether number ρ(d,g,3) is negative, d≥20 and g≤f(d), where f(d)=d3/2/(6·21/2) + lower order terms, there exists a regular component of Hd,g3 (the closure in Hilbℙ3 of the open set parametrizing smooth, connected, non-degenerate curves in ℙ3) with the expected number of moduli. Moreover, examples of the fact that such components are not unique are given.

Fully faithful Fourier-Mukai functors and generic vanishing

Abstract This paper is devoted to extend some well-known facts on the genus of a surface and on the Heegaard genus of a 3-manifold to manifolds of arbitrary dimension. More precisely, we prove that the genus of non-orientable manifolds is always even and we compare the genus of a manifold with the rank of its fundamental group and with the genus of its boundary.

Curve aritmeticamente Buchsbaum su superfici di grado 3 e 4 dello spazio proiettivo

By a theorem of Wahl, the canonically embedded curves which are hyperplane section of K3 surfaces are distinguished by the non-surjectivity of their Wahl map. In this paper we address the problem of distinguishing hyperplane sections of abelian surfaces. The somewhat surprising result is that the Wahl map of such curves is (tendentially) surjective, but their second Wahl map has corank at least 2 (in fact a more precise result is proved).