On codimension two flats in Fermat-type arrangements

TL;DR

This paper investigates Fermat arrangements of codimension 2 flats in projective spaces, revealing algebraic properties and providing counterexamples to expected ideal containment relations.

Contribution

It introduces Fermat arrangements and demonstrates their role as counterexamples to the conjectured containment between symbolic and ordinary powers of ideals.

Findings

Fermat arrangements have specific algebraic properties.

They serve as counterexamples to ideal containment conjectures.

The study advances understanding of algebraic structures in projective geometry.

Abstract

In the present note we study certain arrangements of codimension $2$ flats in projective spaces, we call them "Fermat arrangements". We describe algebraic properties of their defining ideals. In particular, we show that they provide counterexamples to an expected containment relation between ordinary and symbolic powers of homogeneous ideals.