On Localization of Tight Closure in Line-$S_4$ Quartics

TL;DR

This paper presents a new example of a hypersurface over a field of characteristic two where tight closure fails to commute with localization, using innovative tiling and dynamical system techniques.

Contribution

It introduces a novel hypersurface example demonstrating non-commutation of tight closure and localization, expanding understanding of tight closure behavior in algebraic geometry.

Findings

Tight closure does not commute with localization in the constructed example.

Utilizes a tiling argument with Sierpiński triangles and a dynamical system analysis.

Builds on prior work by Brenner and Monsky to deepen the understanding of tight closure properties.

Abstract

Building on work of Brenner and Monsky from 2010 and on a Hilbert-Kunz calculation of Monsky from 1998, we exhibit a novel example of a hypersurface over $\overline{\mathbb{F}_2}$ in which tight closure does not commute with localization. Our methods involve a surprising tiling argument using Sierpi\'nski triangles, as well as an inspection of a certain dynamical system in characteristic two.