Regularity under Integral Closure: A General Criterion and a Sharp Counterexample

Abstract

We study the behavior of Castelnuovo--Mumford regularity under integral closure. We prove a general cohomological criterion for the inequality $\mathrm{reg}(\overline{J})\le reg(J)$, and in particular obtain it for every homogeneous ideal in a polynomial ring in at most three variables. The proof uses a high-degree comparison between $\overline{J}$ and its saturation. We also give a saturated monomial counterexample in four variables, showing that the three-variable result is sharp.

7 pages. Substantially revised. Added a general cohomological criterion and a high-degree agreement result; proved the inequality for all homogeneous ideals in at most three variables