On the computation of Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring

TL;DR

This paper introduces algorithms to compute the Castelnuovo-Mumford regularity of Rees algebras and fiber rings of certain ideals, providing a counter-example to a previous conjecture about their equality.

Contribution

The paper develops algorithms for regularity computation and disproves a conjecture relating the regularities of Rees algebras and fiber rings.

Findings

Algorithms successfully compute regularities for specific ideals.

Counter-example disproves the conjecture that these regularities are always equal.

Provides new insights into the structure of Rees algebras and fiber rings.

Abstract

We present algorithms for the computation of the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring of equigenerated $\mm$-primary ideals in two variables. Applying these algorithms, we find a counter-example to a conjecture of Eisenbud and Ulrich which states that these regularities are equal.