Rees Algebras of Truncations of Complete Intersections

TL;DR

This paper characterizes the defining equations of Rees algebras and special fiber rings for truncations of complete intersection ideals, revealing their Koszul properties and providing explicit generators and relations.

Contribution

It explicitly describes the Rees algebra of truncations of complete intersections using a new approach involving a related Rees ring and computes its Groebner basis.

Findings

R(M) is a Koszul algebra.

R(I) is often a Koszul algebra.

Provides explicit generators and relations for Rees algebras.

Abstract

In this paper we describe the defining equations of the Rees algebra and the special fiber ring of a truncation I of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m. To describe explicitly the Rees algebra R(I) in terms of generators and relations we map another Rees ring R(M) onto it, where M is the direct sum of powers of m. We compute a Groebner basis of the ideal defining R(M). It turns out that the normal domain R(M) is a Koszul algebra and from this we deduce that in many instances R(I) is a Koszul algebra as well.