Castelnuovo-Mumford regularity of unprojections and the Eisenbud-Goto regularity conjecture

TL;DR

This paper introduces a new unprojection-based method to construct counterexamples to the Eisenbud-Goto regularity conjecture for fixed dimensions and codimensions, highlighting the impact of singularities on regularity.

Contribution

It presents a novel unprojection approach to generate counterexamples for the Eisenbud-Goto conjecture across various dimensions and codimensions, emphasizing the role of singularities.

Findings

Counterexamples for fixed dimension and codimension

Unprojection method for constructing regularity counterexamples

Singularities influence on Castelnuovo-Mumford regularity

Abstract

McCullough and Peeva found sequences of counterexamples to the Eisenbud--Goto conjecture on the Castelnuovo--Mumford regularity by using Rees-like algebras, where entries of each sequence have increasing dimensions and codimensions. In this paper we suggest another method to construct counterexamples to the conjecture with any fixed dimension $n\geq3$ and any fixed codimension $e\geq2$. Our strategy is an unprojection process and utilizes the possible complexity of homogeneous ideals with three generators. Furthermore, our counterexamples exhibit how singularities affect the Castelnuovo--Mumford regularity.