Ordinary and symbolic Rees algebras for ideals of Fermat point configurations

TL;DR

This paper provides a detailed algebraic analysis of Fermat ideals, including their generators, resolutions, and Rees algebras, revealing their Noetherian symbolic Rees algebra and regularity properties.

Contribution

It systematically describes the generators and resolutions of Fermat ideals' powers and proves the Noetherian property of their symbolic Rees algebras, advancing understanding of their algebraic structure.

Findings

Symbolic Rees algebras of Fermat ideals are Noetherian.

Explicit formulas for Castelnuovo-Mumford regularity of powers.

Determination of reduction ideals for Fermat ideals.

Abstract

Fermat ideals define planar point configurations that are closely related to the intersection locus of the members of a specific pencil of curves. These ideals have gained recent popularity as counterexamples to some proposed containments between symbolic and ordinary powers. We give a systematic treatment of the family of Fermat ideals, describing explicitly the minimal generators and the minimal free resolutions of all their ordinary powers as well as many symbolic powers. We use these to study the ordinary and the symbolic Rees algebra of Fermat ideals. Specifically, we show that the symbolic Rees algebras of Fermat ideals are Noetherian. Along the way, we give formulas for the Castelnuovo-Mumford regularity of the powers of Fermat ideals and we determine their reduction ideals.