Natural Cohomology on $\mathbb{P}^1 \times \mathbb{P}^1$

TL;DR

This paper proves Eisenbud and Schreyer's conjecture that there exist vector bundles on  imes  with natural cohomology for specific twists and prescribed Hilbert polynomials, advancing understanding of vector bundle cohomology.

Contribution

The paper confirms the existence of vector bundles with natural cohomology on  imes  for given Hilbert polynomials, validating a conjecture in algebraic geometry.

Findings

Confirmed the conjecture of Eisenbud and Schreyer.

Constructed vector bundles with natural cohomology.

Established existence for prescribed Hilbert polynomials.

Abstract

A vector bundle on a projective variety has a natural cohomology if for every twist its cohomology is concentrated in a single degree. Eisenbud and Schreyer conjectured there should be vector bundles on $\mathbb{P}^1 \times \mathbb{P}^1$ with natural cohomology with respect to bundles $\mathcal{O}(1,0),\mathcal{O}(0,1)$ with prescribed Hilbert polynomial. We prove this conjecture.