P-Forms from Syzygies

TL;DR

This paper introduces a method to construct polynomial differential forms that vanish on specific loci using syzygies of homogeneous ideals, with practical implementation in Macaulay2 and applications to instanton bundles.

Contribution

It develops a new tool for constructing differential forms via syzygies and demonstrates its application to instanton bundles in algebraic geometry.

Findings

Successfully implemented the method in Macaulay2.

Constructed examples of instanton bundles of charges 4 and 5.

Provided a practical computational approach for geometric constructions.

Abstract

These notes aim to develop a tool for constructing polynomial differential $p$-forms vanishing on prescribed loci through syzygies of homogeneous ideals. Examples are provided through implementing this method in Macaulay2, particularly examples of instanton bundles of charges $4$ and $5$ on $\mathbb{P}^3$ that arise in this construction.