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

TL;DR

This paper proves a conjecture about the existence of natural vector bundles on the bi-projective plane \\(\\mathbb{P}^1 \\times \\mathbb{P}^1\\) with prescribed Euler characteristics, expanding the duality theory between Betti diagrams and cohomology tables.

Contribution

It establishes the existence of natural vector bundles with specified Euler characteristics on \\(\\mathbb{P}^1 \\times \\mathbb{P}^1\\) under certain conditions, confirming a conjecture by Eisenbud and Schreyer.

Findings

Proved the conjecture for non-integral \\(\\alpha, \\beta\\).

Extended Boij-S"oderberg duality to bi-graded settings.

Characterized conditions for existence of natural vector bundles.

Abstract

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