The ubiquity of Sylvester forms in almost complete intersections

TL;DR

This paper investigates the structure of Rees algebras of almost complete intersection ideals in low-dimensional polynomial rings, highlighting the role of Sylvester forms and their connection to Cohen–Macaulay properties.

Contribution

It introduces a novel approach using Sylvester forms and iterative mapping cones to analyze Rees algebras of specific ideals, including monomials and those related to plane Cremona maps.

Findings

Rees ideals generated by Sylvester forms are almost Cohen–Macaulay.

Identifies classes of ideals where Rees algebra structure is explicitly characterized.

Proves Rees ideals of certain plane Cremona maps are generated by Sylvester forms.

Abstract

The subject matter is the structure of the Rees algebra of almost complete intersection ideals of finite colength in low-dimensional polynomial rings over fields. The main tool is a mix of Sylvester forms and iterative mapping cone construction. The material developed spins around ideals of forms in two or three variables in the search of those classes for which the corresponding Rees ideal is generated by Sylvester forms and is almost Cohen--Macaulay. A main offshoot is in the case where the forms are monomials. Another consequence is a proof that the Rees ideals of the base ideals of certain plane Cremona maps (e.g., de Jonqui\`eres maps) are generated by Sylvester forms and are almost Cohen--Macaulay.