Syzygies of ${\mathbb{P}}^1\times {\mathbb{P}}^1$: data and conjectures

Abstract

We provide a number of new conjectures and questions concerning the syzygies of ${\mathbb{P}}^1\times {\mathbb{P}}^1$. The conjectures are based on computing the graded Betti tables and related data for large number of different embeddings of ${\mathbb{P}}^1\times {\mathbb{P}}^1$. These computations utilize linear algebra over finite fields and high-performance computing.