The integral closure of a primary ideal is not always primary

TL;DR

This paper provides counterexamples showing that the integral closure of a primary ideal is not always primary, explores Whitney and Zariski equisingularity conditions, and compares different stratifications of a specific hypersurface.

Contribution

It extends Krull's question by providing counterexamples across all characteristics and analyzes the stratification differences of a notable hypersurface.

Findings

Counterexamples to Krull's question in polynomial rings of any characteristic.

The Jacobian ideal of a specific polynomial is a counterexample to primary ideal closure.

Whitney stratification differs from isosingular set stratification for the studied hypersurface.

Abstract

In 1936, Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic $p=2$. We provide counterexamples to Krull's question regarding polynomial rings with any characteristics. We also find that the Jacobian ideal $J$ of the polynomial $f = x^6 + y^6 + x^4 z t + z^3$ given by Brian\c{c}on and Speder in 1975 is a counterexample to Krull's question. Let $V_1$ be the hypersurface defined by $f = 0$ and $V_2$ be its singular locus. Brian\c{c}on and Speder proved that Whitney equisingularity does not imply Zariski equisingularity by showing that the pair $(V_1 \setminus V_2,\ V_2)$ satisfies Whitney's conditions around the origin but fails Zariski's equisingular conditions. We discover that the pair $(V_1 \setminus V_2,\ V_2)$ fails Whitney's conditions at the variety of the embedded prime of the integral closure $\bar{J}$, which means that $V_1$ is not Whitney regular along $V_2$. Moreover, we also show that Whitney stratification of this hypersurface is different from the stratification of isosingular sets given by Hauenstein and Wampler, which is related to Thom-Boardman singularity.