Counterexamples to the $I^{(3)} \subset I^2$ containment

TL;DR

This paper demonstrates that, contrary to previous assumptions, the third symbolic power of certain radical ideals in the complex projective plane is not contained within their second ordinary power, providing a negative answer to a longstanding question.

Contribution

It provides the first known counterexamples to the containment $I^{(3)} ot hinspace extsubseteq hinspace I^2$ for radical ideals of points in the complex projective plane.

Findings

Counterexamples to the containment $I^{(3)} ot hinspace extsubseteq hinspace I^2$

Answers negatively a question by Huneke and Harbourne

Shows the non-containment in general cases

Abstract

We show that in general the third symbolic power of a radical ideal of points in the complex projective plane is not contained in the second usual power of that ideal. This answers in negative a question asked by Huneke and generalized by Harbourne.