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

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.