The role of the cotangent bundle in resolving ideals of fat points in the plane

TL;DR

This paper explores how the cotangent bundle influences the algebraic structure of fat point schemes in the plane, proposing conjectures and providing partial evidence for their validity.

Contribution

It introduces two new conjectures linking cotangent bundle properties to fat point schemes, supported by examples and partial results.

Findings

Proposed two conjectures relating cotangent bundle to fat point schemes

Provided examples supporting the conjectures

Presented partial results indicating potential validity

Abstract

We study the connection between the generation of a fat point scheme supported at general points in the plane and the behaviour of the cotangent bundle with respect to some rational curves particularly relevant for the scheme. We put forward two conjectures, giving examples and partial results in support of them.