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

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 $$\mathbb{P}$^1 \times $\mathbb{P}$^1$withnaturalcohomologywithrespecttobundles$$\mathcal{O}$(1,0),$\mathcal{O}$(0,1)$ with prescribed Hilbert polynomial. We prove this conjecture.