Horrocks Correspondence on ACM Varieties

TL;DR

This paper generalizes Horrocks correspondence for ACM varieties by describing vector bundles through cohomological invariants and socle modules, providing existence and uniqueness theorems, with detailed cases for quadrics and Veronese surfaces.

Contribution

It extends Horrocks correspondence to a broader class of ACM varieties using cohomological and socle invariants, with constructive and uniqueness results.

Findings

Constructed vector bundles from cohomological invariants.

Proved that invariants determine bundles up to isomorphism.

Analyzed specific cases of quadrics and Veronese surfaces.

Abstract

We describe a vector bundle $\sE$ on a smooth $n$-dimensional ACM variety in terms of its cohomological invariants $H^i_*(\sE)$, $1\leq i \leq n-1$, and certain graded modules of "socle elements" built from $\sE$. In this way we give a generalization of the Horrocks correspondence. We prove existence theorems where we construct vector bundles from these invariants and uniqueness theorems where we show that these data determine a bundle up to isomorphisms. The cases of the quadric hypersurface in $\mathbb P^{n+1}$ and the Veronese surface in $\mathbb P^5$ are considered in more detail.