Segre's Regularity Bound for Fat Point Schemes

TL;DR

This paper proves the Segre's conjectured upper bound for the Castelnuovo-Mumford regularity of fat point schemes, providing a new partition result for matroids and an improved regularity bound in certain cases.

Contribution

It confirms the Segre's conjecture on regularity bounds and introduces a new matroid partition result that refines existing bounds in algebraic geometry.

Findings

Segre's regularity bound conjecture is proven to be true.

An alternative regularity bound that improves Segre's bound in some cases.

A new partition theorem for matroids is established.

Abstract

Motivated by questions in interpolation theory and on linear systems of rational varieties, one is interested in upper bounds for the Castelnuovo-Mumford regularity of arbitrary subschemes of fat points. An optimal upper bound, named after Segre, was conjectured by Trung and, independently, by Fattabi and Lorenzini. It is shown that this conjecture is true. Furthermore, an alternate regularity bound is established that improves the Segre bound in some cases. Among the arguments is a new partition result for matroids.