On the regularity index of $s$ equimultiple fat points not on a linear $(r-1)$-space, $s \le r+3$

TL;DR

This paper proves Trung's conjecture regarding Segre's upper bound for a specific configuration of equimultiple fat points in algebraic geometry, using algebraic methods that could apply to other cases.

Contribution

It confirms Trung's conjecture for s ≤ r+3, providing a new algebraic approach that can be extended to other configurations of fat points.

Findings

Proved Trung's conjecture for s ≤ r+3.

Validated Segre's upper bound in this context.

Introduced an algebraic method applicable to other fat point cases.

Abstract

We prove the Trung's conjecture about Segre's upper bound for s equimultiple fat points not on a linear (r-1)-space, s\le r+3, by algebraic method used in [3]. This method also may used to research other cases of fat points.