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Giuseppe Pareschi

Publications and source records attributed to Giuseppe Pareschi.

At least 19 recordsLinked to original sources
arXiv2026

Abelian varieties analogs of two results about algebraic curves

We characterize decomposable principally polarized abelian varieties of the form $E\times B$, with $E$ an elliptic curve, in two different ways, which are, surprisingly, completely analogous to classical results of curve theory concerning hyperelliptic curves. The first one is by the failure of a normal generation property, namely the generation in degree zero of a certain graded module over the symmetric algebra over $H^0(2Θ)$. This appears to be the first result of this type in the realm of p.p.a.v.'s. The second characterization is by the failure of surjectivity of second order gaussian maps associated to line bundles corresponding to $6Θ$, or, equivalently, by the fact that at some point, the line bundle corresponding to $3Θ$ fails to separate $2$-jets. We also show that this last result is equivalent to an effective version of a theorem of Nakamaye characterizing the above decomposable abelian varieties as those computing the minimal Seshadri constant. Finally we propose some conjectural generalizations relating $p$-jets separation thresholds, higher gaussian maps sujectivity thresholds, and Seshadri constants.

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arXiv2026

Singularities of base loci on abelian varieties

We prove that the log canonical threshold of the base ideal of a complete linear system on a complex abelian variety is $\ge 1$, and equality holds if and only if the base locus has divisorial components. Consequently the same assertions hold for the ideal of the intersection of translates of theta divisors by the points of a finite subgroup.

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arXiv2025

Semihomogenous vector bundles, $\mathbb Q$-twisted sheaves, duality, and linear systems on abelian varieties

In this paper we point out the natural relation between $\mathbb Q$-twisted objects of the derived category of abelian varieties, cohomological rank functions, and semihomogeneous vector bundles. We apply this to two basic classes of objects, corresponding to each other via the Fourier-Mukai-Poincaré transform: positive twists of the ideal sheaf of one point and of the evaluation complexes of ample simple semihomogeneous vector bundles. This naturally leads to the introduction of $\mathbb Q^{\ge 0}$- graded section modules associated to line bundles on abelian varieties built by means of semihomogeneous vector bundles (containing the usual section rings). We prove a duality relation between such modules associated to dual polarizations, which is not visible at the level of the usual section rings. Other applications include formulas relating the thresholds of relevant cohomological rank functions appearing in this context. As a consequence we show a lower bound for the base point free threshold of a polarization in function of its type, and some obstructions to surjectivity of multiplication maps of global sections of certain line bundles.

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arXiv2025

Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: an algebro-geometric approach

This paper is withdrawn since we found a flaw in the proof of Theorem 4, asserting that the base locus of the complete linear system of an ample line bundle on a complex abelian variety is reduced. The error is in page 7, line $ -14$, where we claim that the divisor "mathcal E" on the variety $X$ is linearly equivalent to zero. This is untrue. For instance, it would imply that, for a non-torsion point $x$ on an abelian surface $A$, letting $E_x$, $E_{-x}$, and $E_0$ the exceptional curves in the blow up of $A$ at $x$, $-x$, and $0$, then $2E_0$ is linearly equivalent to $E_x +E_{-x}$, which is easily seen to be false. Therefore Theorem 4 of our paper has to be considered unproven. We still believe that it holds true. All the other arguments of our paper are correct but unfortunately they depend on the above mentioned Theorem 4. To be precise, from Theorem 4 follows Theorem 3, asserting that the scheme $Σ(X,Θ, G)$ of Definition 9 is reduced. The rest of the paper contains algebro-geometric proofs of Shiota's theorem characterizing Jacobians via the KP equation (Section 4), and of Krichever's theorems characterizing Jacobians by the existence of an inflectionary or degenerate trisecant to the Kummer variety embedded in $\mathbb P^{2^g-1}$ (Theorem 18 and Theorem 25). Also these proofs are correct but they depend on a weaker version of Theorem 3, namely on the assertion that the components of codimension two of the scheme $Σ(X,Θ, G)$ are generically reduced. In turn, this weaker version of Theorem 3 would follow, by the same argument used in its proof, from a conjecture by Debarre asserting that the base locus of the complete linear system of an ample line bundle on an abelian variety is generically reduced in codimension two.

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arXiv2023

Derived invariance of the Albanese relative canonical ring

We show the derived invariance of various geometric invariants of smooth complex projective varieties governed by the Albanese map, including the relative canonical ring and the class of the relative canonical model in a suitable variant of the Grothendieck ring of varieties. Then we derive some applications to the derived invariance of Hodge numbers.

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arXiv2023

Generation and ampleness of coherent sheaves on abelian varieties, with application to Brill-Noether theory

We introduce a variant of global generation for coherent sheaves on abelian varieties which, under certain circumstances, implies ampleness. This extends a criterion of Debarre asserting that a continuously globally generated coherent sheaf on an abelian variety is ample. We apply this to show the ampleness of certain sheaves, which we call naive Fourier-Mukai-Poincaré transforms, and to study the structure of GV sheaves. In turn, one of these applications allows to extend the classical existence and connectedness results of Brill-Noether theory to a wider context, e.g. singular curves equipped with a suitable morphism to an abelian variety. Another application is a general inequality of Brill-Noether type involving the Euler characteristic and the homological dimension.

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arXiv2021

Torsion points on theta divisors and semihomogeneous vector bundles

We generalize to $n$-torsion a result of Kempf's describing $2$-torsion points lying on a theta divisor. This is accomplished by means of certain semihomogeneous vector bundles introduced and studied by Mukai and Oprea. As an application, we prove a sharp upper bound for the number of $n$-torsion points on a theta divisor, and show that this is achieved only in the case of products of elliptic curves, settling in the affirmative a conjecture of Auffarth, Pirola and Salvati Manni.

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arXiv2020

Singularities of divisors of low degree on simple abelian varieties

It is known by results of Kollár, Ein, Lazarsfeld, Hacon and Debarre that divisors representing principal and other low degree polarizations on abelian varieties have mild singularities. In this note we extend such results to polarizations of degree $<g$ on simple $g$-dimensional abelian varieties, settling a conjecture of Debarre and Hacon.

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arXiv2020

2-torsion points on theta divisors

In this note we prove a sharp bound for the number of 2-torsion points on a theta divisor and show that this is achieved only in the case of products of elliptic curves. This settles in the affirmative a conjecture of Marcucci and Pirola.

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arXiv2018

Cohomological rank functions on abelian varieties

Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of $\mathbb Q$-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincaré transform, which has several consequences. In many concrete geometric contexts these functions provide useful invariants. We illustrate this with two different applications, the first one to GV-subschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties.

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arXiv2018

Derived invariants arising from the Albanese map

Let $a_X:X\rightarrow \mathrm{Alb}\, X$ be the Albanese map of a smooth complex projective variety. Roughly speaking in this note we prove that for all $i \geq 0$ and $α\in \mathrm{Pic}^0\, X$, the cohomology ranks $h^i(\mathrm{Alb}\, X, \,{a_X}_* ω_X\otimes P_α)$ are derived invariants. In the case of varieties of maximal Albanese dimension this proves conjectures of Popa and Lombardi-Popa -including the derived invariance of the Hodge numbers $h^{0,j}$ -- and a weaker version of them for arbitrary varieties. Finally we provide an application to derived invariance of certain irregular fibrations.

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arXiv2017

Torsion points on theta divisors

Using the irreducibility of a natural irreducible representation of the theta group of an ample line bundle on an abelian variety, we derive a bound for the number of $n$-torsion points that lie on a given theta divisor. We present also two alternate approaches to attacking the case $n=2$.

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arXiv2016

Standard canonical support loci

We consider the union of certain irreducible components of cohomological support loci of the canonical bundle, which we call standard. We prove a structure theorem about them and single out some particular cases, recovering and improving results of Beauville and Chen-Jiang. Finally, as an example of application, we extend to compact Kahler manifolds the classification of smooth complex projective varieties with $p_1(X)=1$, $p_3(X)=2$ and $q(X)=\dim X$.

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arXiv2016

Hodge modules on complex tori and generic vanishing for compact Kähler manifolds

We extend the results of generic vanishing theory to polarizable real Hodge modules on compact complex tori, and from there to arbitrary compact Kähler manifolds. As applications, we obtain a bimeromorphic characterization of compact complex tori among compact Kähler manifolds, semi-positivity results, and a description of the Leray filtration for maps to tori.

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arXiv2016

Fully faithful Fourier-Mukai functors and generic vanishing

The aim of this mainly expository note is to point out that, given an Fourier-Mukai functor, the condition making it fully faithful is an instance of \emph{generic vanishing}. We test this point of view on some fairly classical examples, including the strong simplicity criterion of Bondal and Orlov, the standard flip and the Mukai flop.

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arXiv2014

Gaussian maps and generic vanishing I: subvarieties of abelian varieties

We present an approach to Green-Lazarsfeld's generic vanishing combining gaussian maps and the Fourier-Mukai transform associated to the Poincarè line bundle. As an application we prove the Generic Vanishing Theorem for all normal Cohen-Macaulay subvarieties of abelian varieties over an algebraically closed field. This paper is a reworking of part of arXiv:math/0310026v2 [math.AG], which contained a mistake.

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arXiv2012

Generic vanishing, gaussian maps, and Fourier-Mukai transform

In the first part of this paper we prove a vanishing criterion for higher direct images of projective families of line bundles on a Cohen-Macaulay variety X. The result involves certain first-order deformations of certain curves on X, and makes essential use of the notion of global co-gaussian maps, a generalization of Wahl's gaussian maps. In the second part we apply the criterion above, combined with Fourier-Mukai transform on abelian varieties, to prove an algebraic version of Green-Lazarsfeld's Generic Vanishing Theorem. In fact we prove a stronger result concerning higher direct images of Poincaré line bundles, which -- in the compact Kähler setting -- was conjectured by Green and Lazarsfeld and was recently proved, by completely different methods, by Hacon (math.AG/0308198)

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