Related papers: Syzygies of Kummer varieties
Let $X$ be a smooth projective geometrically connected variety defined over a number field $K$. We prove that the geometric \'etale cohomology of $X$ with $\mathbb{Q}/\mathbb{Z}$-coefficients has finitely many classes invariant under the…
In this Note we prove the vanishing of (twisted) Koszul cohomology groups $K_{p,1}$ of an abelian variety with values in powers of an ample line bundle. It complements the work of G. Pareschi on the property $(N_p)$.
This paper studies the structure of the Mordell--Weil groups of semiabelian varieties over algebraic extensions of number fields whose absolute Galois group is finitely generated, with particular emphasis on that generated by a single…
The rational homology groups of the packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre--Veronese varieties). These complexes…
We study the structure of the Mordell--Weil groups of semiabelian varieties over large algebraic extensions of a finitely generated field of characteristic zero. We consider two types of algebraic extensions in this paper; one is of…
Following an article of Dettweiler and Sabbah, this article studies the behaviour of various Hodge invariants by middle additive convolution with a Kummer module. The main result gives the behaviour of the nearby cycle local Hodge numerical…
We prove several results on the number of rational points on open subsets of Kummer varieties of arbitrary dimension. Some of our results are unconditional, and others depend on the Parity Conjecture (a corollary of the Conjecture of Birch…
It is proved, as was conjectured by Eisenbud-Koh-Stillman, that for a finitely generated graded module $M$ over the symmetric algebra $S(V)$, if the Koszul group ${\cal K}_{p,0}(M,V)\ne 0$, then the set of rank 1 relations in $M_0\otimes V$…
Finding the maximal dimension of complete subvarieties of the moduli space of smooth $n$-pointed curves of genus $g$ is a long-standing open problem. Here we show that for $g\ge 3\cdot 2^{d-1}$, if the characteristic of the base field is…
The present paper is related to a conjecture made by Green and Lazarsfeld concerning 1-linear syzygies of curves embedded by complete linear systems of sufficiently large degrees. Given a smooth, irreducible, complex, projective curve $X$,…
This paper studies syzygies of curves that have been embedded in projective space by line bundles of large degree. The proofs take advantage of the relationship between syzygies and spaces of section of vector bundles associated to the…
Generalizing the well-known Green Conjecture on syzygies of canonical curves, Green and Lazarsfeld formulated in 1986 the Secant Conjecture predicting that a line bundle L of sufficiently high degree on a curve has a non-linear p-syzygy if…
In this paper, we study asymptotic behavior of projective embeddings of Kummer varieties given by theta functions, and their amoebas. We prove that a Lagrangian fibration of the Kummer variety can be approximated by moment maps of the…
In char $k = p >0$, A. Langer proved a strong restriction theorem (in the style of H. Flenner) for semistable sheaves to a very general hypersurface of degree $d$, on certain varieties, with the condition that `char $k > d$'. He remarked…
We survey results concerning behavior of positivity of line bundles and possible vanishing theorems in positive characteristic. We also try to describe variation of positivity in mixed characteristic. These problems are very much related to…
Deciding the presence of the weak Lefschetz property often is a challenging problem. In this work an in-depth study is carried out in the case of Artinian monomial ideals with four generators in three variables. We use a connection to…
The goal of this article is to study the equations and syzygies of embeddings of rational surfaces and certain Fano varieties. Given a rational surface X and an ample and base-point-free line bundle L on X, we give an optimal numerical…
In this paper, we investigate the stabilizers of certain multisets $\mod p^k$ with respect to their natural multiplicative action, completely describing them for a certain family of polynomials whenever $p$ is an odd prime. This elucidates…
We partially generalize the theory of semihomogeneous bundles on an abelian variety $A$ developed by Mukai. This involves considering abelian subvarieties $Y\subset X_A=A\times\hat{A}$ and studying coherent sheaves on $A$ invariant under…
We study the Ein-Lazarsfeld Conjecture of syzygies of Veronese varieties in the case of linear syzygies q=1. We show a vanishing statement which agrees with the conjecture up to highest and second-highest order for linear syzygies.