Locally Cohen-Macaulay space curves defined by cubic equations and globally generated vector bundles

TL;DR

This paper classifies certain globally generated vector bundles on projective spaces by analyzing locally Cohen-Macaulay space curves defined by cubic equations, extending previous results to higher dimensions and providing explicit resolutions.

Contribution

It introduces a classification approach for globally generated vector bundles with specific Chern classes using monads of Cohen-Macaulay curves, extending to higher-dimensional projective spaces.

Findings

Classification of globally generated vector bundles with c1 ≥ 4 on P^3.

Extension of classification to higher-dimensional projective spaces.

Explicit graded free resolutions for certain space curves.

Abstract

We classify globally generated vector bundles with first Chern class $c_1$ at least 4 on the projective 3-space with the property that $E(-c_1+3)$ has a non-zero global section. This (seemingly) technical result allows one to reduce the classification of globally generated vector bundles with $c_1$ at most 7 on the projective 3-space to the classification of stable rank-2 reflexive sheaves with the same properties. The proof is based on a description of the monads of all locally Cohen-Macaulay space curves defined by cubic equations. We extend then this kind of classification to higher dimensional projective spaces. We use this extension to recuperate quickly the classification of globally generated vector bundles with $c_1=4$ on the projective $n$-space for $n$ at least 4, which is part of the main result of our previous paper [arxiv:1305.3464]. We provide, in the appendices to the paper, graded free resolutions for the homogeneous ideals and for the graded structural algebras of all non-reduced locally Cohen-Macaulay space curves of degree at most 4.