# Singularities of Rees-like Algebras

## TL;DR

This paper investigates the singularities of Rees-like algebras, introduces prime standardization to control singular locus size, and demonstrates that certain hyperplane sections satisfy the Eisenbud-Goto Conjecture, revealing new structural properties.

## Contribution

It develops prime standardization to preserve singular locus codimension and analyzes the regularity and Cohen-Macaulay properties of Rees-like algebras.

## Key findings

- The singular locus codimension can increase under homogenization.
- Certain hyperplane sections of Rees-like algebras satisfy the Eisenbud-Goto Conjecture.
- Rees-like algebras are always Cohen-Macaulay and have a self-dual canonical module.

## Abstract

Recently, Peeva and the second author constructed irreducible projective varieties with regularity much larger than their degree, yielding counterexamples to the Eisenbud-Goto Conjecture. Their construction involved two new ideas: Rees-like algebras and step-by-step homogenization. Yet, all of these varieties are singular and the nature of the geometry of these projective varieties was left open. The purpose of this paper is to study the singularities inherent in this process. We compute the codimension of the singular locus of an arbitrary Rees-like algebra over a polynomial ring. We then show that the relative size of the singular locus can increase under step-by-step homogenization. To address this defect, we construct a new process, we call prime standardization, which plays a similar role as step-by-step homogenization but also preserves the codimension of the singular locus. This is derived from ideas of Ananyan and Hochster and we use this to study the regularity of certain smooth hyperplane sections of Rees-like algebras, showing that they all satisfy the Eisenbud-Goto Conjecture, as expected. On a more qualitative note, while Rees-like algebras are almost never Cohen-Macaulay and never normal, we characterize when they are seminormal, weakly normal, and, in positive characteristic, F-split. Finally, we construct a finite free resolution of the canonical module of a Rees-like Algebra over the presenting polynomial ring showing that it is always Cohen-Macaulay and has a surprising self-dual structure.