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Boij-S\"oderberg theory concerns resolutions of graded modules over apolynomial ring over a field. Specifically Boij-S\"oderberg theory gives adescription of the cone of Betti diagrams for Cohen-Macaulay modules. Eisenbudand Schreyer discovered a duality between the cone of Betti diagrams and thecone of cohomology tables for vector bundles on projective space. In the dualtheory an important role is played by so called natural vector bundles $E$which have the property that the cohomology of every twist of $E$ isconcentrated in a single degree. In [4], Eisenbud and Schreyer consider thebi-graded theory on $\mathbb{P}^1 \times \mathbb{P}^1$ and conjecture thatnatural vector bundles exist with prescribed Euler characteristic. The Eulercharacterist depends on three rational number $\alpha,\beta, \gamma$. We provethis conjecture provided that $\alpha,\beta$ are not both integral.

Hunting Vector Bundles on $\mathbb{P}^1 \times \mathbb{P}^1$