Boij–Söderberg Theory

Abstract

In this article we start with a survey of the recent breakthroughs concerning Betti table of graded modules over the polynomial ring and cohomology tables coherent sheaves on projective space. We then ask how this theory can be extended to a more general setting. Our first new result concerns cohomology tables of very ample polarized varieties. We prove a necessary and sufficient condition that the Boij–Söderberg cone of cohomology tables of coherent sheaves on a variety X of dimension d with polarization $\mathcal{O}_{X}(1)$ coincides with the corresponding cone of $(\mathbb{P}^{d}, \mathcal{O}(1))$, and conjecture that our condition is always satisfied. The last section concerns cohomology tables of vector bundles with respect to more than one line bundle, where we start with the simplest case ℙ¹×ℙ¹. We identify some extremal rays in the Boij–Söderberg cone of vector bundles on ℙ¹×ℙ¹, and conjecture that these are all.