A structure theorem for syzygies of del Pezzo varieties

TL;DR

This paper provides a structural description of the highest linear syzygies among quadrics defining del Pezzo varieties, using the Buchsbaum-Eisenbud theorem to represent these syzygies via skew-symmetric matrices.

Contribution

It introduces a new structure theorem for syzygies of del Pezzo varieties, linking them to skew-symmetric matrices and wedge products of linear forms.

Findings

Syzygies can be represented as columns of skew-symmetric matrices.

The structure theorem applies to the highest linear syzygies among quadrics.

Provides a new perspective on the algebraic structure of del Pezzo varieties.

Abstract

Using the Buchsbaum-Eisenbud structure theorem for a minimal free resolution of an arithmetically Gorenstein variety, we describe a structure theorem for the highest linear syzygies among quadrics defining a del Pezzo variety. Indeed, such syzygies can be represented as columns of a skew-symmetric matrix whose entries are wedge products of linear forms.