On syzygies of Segre embeddings of $P^1\times P^1$

TL;DR

This paper constructs a nonminimal free resolution for Segre embeddings of P^1×P^1, providing explicit formulas for specific Betti table entries, enhancing understanding of their syzygies over any field.

Contribution

It introduces a nonminimal graded free resolution for Segre embeddings of P^1×P^1 and derives explicit formulas for Betti table entries, advancing syzygy analysis.

Findings

Explicit formulas for Betti table entries at the end of the first row.

Construction of a nonminimal graded free resolution.

Results valid over any field.

Abstract

We construct a nonminimal graded free resolution of Segre embeddings of $P^1\times P^1$, although we don't compute all maps. We use this to prove an explicit formula for certain nonzero entries in the graded Betti table, at the end of the first row. We work over an arbitrary field $k$.