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

TL;DR

This paper investigates the syzygies of the product of two projective lines, providing new conjectures based on extensive computational data of Betti tables using advanced linear algebra and high-performance computing.

Contribution

It introduces new conjectures about the syzygies of bP^1b bP^1 based on large-scale computational experiments and data analysis.

Findings

Computed extensive Betti tables for various embeddings

Formulated new conjectures on syzygy patterns

Utilized high-performance computing for algebraic geometry data

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.