Castelnuovo theory and the geometric Schottky problem

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.Comment: 17 pages; final version, to appear in J. Reine Angew. Math.; no significant changes, only some small corrections and additions with respect to the previous versio

Castelnuovo-Mumford regularity: Examples of curves and surfaces

The behaviour of Castelnuovo-Mumford regularity under ``geometric'' transformations is not well understood. In this paper we are concerned with examples which will shed some light on certain questions concerning this behaviour

Projective schemes: What is Computable in low degree?

This article first presents two examples of algorithms that extracts information on scheme out of its defining equations. We also give a review on the notion of Castelnuovo-Mumford regularity, its main properties (in particular its relation to computational issues) and different ways that were used to estimate it