A counterexample to a conjecture of K\"uronya and Pintye on regularity and integral closure

TL;DR

The paper presents a specific monomial ideal counterexample where the regularity of its integral closure exceeds that of the ideal itself, challenging a conjecture by K"uronya and Pintye.

Contribution

It provides the first explicit counterexample to the polynomial-ring formulation of the K"uronya--Pintye conjecture using monomial ideals.

Findings

The ideal I is generated in degree 4 with reg(I)=4.

The integral closure of I has a generator of degree 5 with reg(rac{I})=5.

This counterexample disproves the conjecture in the polynomial ring setting.

Abstract

We exhibit an equigenerated monomial ideal $I\subseteq K[x,y,z,w]$ with $\operatorname{reg}(\overline{I})>\operatorname{reg}(I)$. The ideal $I$ is generated in degree 4 and satisfies $\operatorname{reg}(I)=4$, while its integral closure $\overline{I}$ has a minimal generator of degree 5 and satisfies $\operatorname{reg}(\overline{I})=5$. This gives a counterexample to the polynomial-ring formulation of the K\"uronya--Pintye conjecture.