Moduli of Very Ample Line Bundles

TL;DR

This paper constructs a moduli space for very ample line bundles on projective varieties, introduces generalized Fitting ideals for complexes of sheaves, and applies these to Brill-Noether spaces and loci with higher projective dimension.

Contribution

It develops a new framework for moduli of line bundles and generalizes Fitting ideals to complexes, enabling new geometric and scheme-theoretic applications.

Findings

Constructed a moduli space of very ample line bundles.

Generalized Fitting ideals to complexes of sheaves.

Applied these to Brill-Noether spaces and loci with jump in projective dimension.

Abstract

Let $X$ be a projective variety over a field. In this paper, we will construct a moduli space of very ample line bundles on $X$. In doing so, we develop a generalization of Fitting ideals to complexes of sheaves on $X$. We give other applications of these Fitting ideals such as constructing Brill-Noether spaces for higher dimensional varieties and giving a scheme structure to the locus where the projective dimension of a module jumps up.