On invariant of the regularity index of fat points

TL;DR

This paper investigates the invariance of the regularity index of fat points under linear subspace changes and explores bounds related to non-degenerate equimultiple fat points in projective space.

Contribution

It proves the invariance of the regularity index under certain geometric transformations and characterizes when Segre's bound is attained for specific fat point configurations.

Findings

Regularity index is invariant under changes of the containing linear subspace.

Segre's bound is attained by any set of s non-degenerate equimultiple fat points in P^n, for s ≤ n+3.

Existence of configurations where Segre's bound is not attained for s = n+4 in P^n.

Abstract

We prove invariant of the regularity index of fat points under changes of the linear subspace containing the support of the fat points. Then we show that Segre's bound is attained by any set of s non-degenerate equimultiple fat points in $\mathbb P^n$, $s\le n+3$. We also give an example showing that there always exists a set of n+4 non-degenerate equimultiple fat points in $\mathbb P^n$ such that Segre's bound is not attained.