Horrocks Correspondence on a Quadric Surface

TL;DR

This paper extends the Horrocks correspondence to vector bundles on a quadric surface, establishing a bijection between certain invariants and isomorphism classes of bundles without line summands.

Contribution

It introduces invariants for vector bundles on a product of two projective lines and proves a one-to-one correspondence with their isomorphism classes.

Findings

Established a new correspondence for vector bundles on a quadric surface.

Defined a set of invariants including the first cohomology module.

Proved the bijection between invariants and isomorphism classes.

Abstract

We extend the Horrocks correspondence between vector bundles and cohomology modules on the projective plane to the product of two projective lines. We introduce a set of invariants for a vector bundle on the product of two projective lines, which includes the first cohomology module of the bundle, and prove that there is a one to one correspondence between these sets of invariants and isomorphism classes of vector bundles without line bundle summands.