Extremal Rees Algebras

TL;DR

This paper investigates extremal Rees algebras arising from almost complete intersection ideals, focusing on their nonlinear relations and providing methods to compute their invariants.

Contribution

It introduces new techniques for analyzing extremal Rees algebras, especially those leading to almost Cohen--Macaulay structures, with a focus on nonlinear relations.

Findings

Identification of classes of extremal Rees algebras with combinatorial origins

Development of effective methods to compute algebra invariants

Insights into the structure of almost Cohen--Macaulay Rees algebras

Abstract

We study almost complete intersections ideals whose Rees algebras are extremal in the sense that some of their fundamental metrics---depth or relation type---have maximal or minimal values in the class. The focus is on those ideals that lead to almost Cohen--Macaulay algebras and our treatment is wholly concentrated on the nonlinear relations of the algebras. Several classes of such algebras are presented, some of a combinatorial origin. We offer a different prism to look at these questions with accompanying techniques. The main results are effective methods to calculate the invariants of these algebras.