Blowups of hypersurfaces

TL;DR

This paper investigates the structure of Rees rings of maximal ideals in hypersurface rings, providing minimal generators, Cohen-Macaulayness criteria, and other invariants, extending classical results on regular local rings.

Contribution

It offers a detailed analysis of the Rees ring for hypersurface rings, including explicit generators and homological properties, which was previously not well-understood.

Findings

Minimal generating set for the Rees ring of hypersurface rings

Criteria for Cohen-Macaulayness of the Rees ring

Identification of key invariants of the Rees algebra

Abstract

A classical result of Micali asserts that a Noetherian local ring is regular if and only if the Rees algebra of its maximal ideal is defined by an ideal of linear forms. In this case, this defining ideal may be realized as a determinantal ideal of generic height, and so the Rees ring is easily resolved by the Eagon-Northcott complex, providing a wealth of information. If $R$ is a non-regular local ring, it is interesting to ask how far the Rees ring of its maximal ideal strays from this form, and whether any homological data can be recovered. In this paper, we answer this question for hypersurface rings, and provide a minimal generating set for the defining ideal of the Rees ring. Furthermore, we determine the Cohen-Macaulayness of this algebra, along with several other invariants.