Graded basic elements

TL;DR

This paper extends the existence theorem for basic elements from affine to quasi-projective cases and explores geometric applications, including degeneracy loci of sections in smooth projective varieties.

Contribution

It generalizes the existence theorem for basic elements to quasi-projective varieties and applies it to geometric problems like degeneracy loci.

Findings

Existence of basic elements in quasi-projective cases proven.

Every local complete intersection of codimension two can be realized as a degeneracy locus.

Applications to the geometry of smooth projective varieties.

Abstract

We prove the existence theorem for basic elements in the quasi-projective case, extending results of Eisenbud-Evans and Bruns from the affine case. We give several geometric applications. For example, we show that every local complete intersection of pure codimension two in a smooth projective variety of dimension d over an infinite field is the degeneracy locus of d-1 sections of a rank d bundle.