We provide a bound on the Θ-regularity of an arbitrary reduced and irreducible curve embedded in a polarized abelian variety in terms of its degree and codimension. This is an “abelian” version of Gruson–Lazarsfeld–Peskine’s bound on the Castelnuovo–Mumford regularity of a non-degenerate curve embedded in a projective space. As an application, we provide a Castelnuovo type bound for the genus o...

The aim of this paper is to show that Castelnuovo theory in projective space (cf. [ACGH] Ch.III §2 and [GH] Ch.4 §3) has a precise analogue for abelian varieties. This can be quite surprisingly related in a concrete way to the geometric Schottky problem, namely the problem of identifying Jacobians among all principally polarized abelian varieties (ppav’s) via geometric conditions on the polariz...

Let $$f:X\rightarrow Y$$ be a semistable fibration between smooth complex varieties of dimension n and m. This paper contains an analysis the local systems de Rham closed relative one forms top on fibers. In particular latter recovers system second Fujita decomposition $$f_*\omega _{X/Y}$$ over higher dimensional base. The so called theory Massey products allows, under natural Castelnuovo-type ...

The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on n-dimensional smooth projective varieties X with an n-block collection B which generates the bounded derived category Db(OX -mod). To this end, we use the theory of n-blocks and Beilinson type spectral sequence to define the notion of regularity of a cohere...

The paper begins by overviewing the basic facts on geometric exceptional collections. Then, we derive, for any coherent sheaf F on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to F and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to co...

We prove asymptotic linear bounds for the Castelnuovo-Mumford regularity of certain filtrations of homogeneous ideals whose Rees algebras need not be Noetherian.