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A counterexample to a conjecture of Küronya and Pintye on regularity and integral closure · arXivDesk

2605.13879May 11, 20263 pages; simplified counterexample with 4 generators instead of 6, main result unchanged

A counterexample to a conjecture of Küronya and Pintye on regularity and integral closure

Soumyadeep Misra

Abstract

We exhibit an equigenerated monomial ideal $I\subseteq K[x,y,z,w]$ with $\operatorname{reg}(\overline{I})>\operatorname{reg}(I)$

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We exhibit an equigenerated monomial ideal I⊆ K[x,y,z,w] with reg(I)>reg(I). The ideal I is generated in degree 4 and satisfies reg(I)=4, while its integral closure I has a minimal generator of degree 5 and satisfies reg(I)=5.

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The paper addresses: A counterexample to a conjecture of Küronya and Pintye on regularity and integral closure.
We exhibit an equigenerated monomial ideal I⊆ K[x,y,z,w] with reg(I)>reg(I).

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Author comment: 3 pages; simplified counterexample with 4 generators instead of 6, main result unchanged

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. 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 Küronya--Pintye conjecture.

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