The equations of Rees algebras of ideals of almost-linear type

TL;DR

This paper provides a comprehensive description of the equations defining Rees algebras of ideals of almost-linear type, extending known results and exploring properties of the canonical morphism in this context.

Contribution

It offers a full characterization of Rees algebra equations for ideals of the form (J,y) with J of linear type, and proves a propagation property for the injectivity of a component of the canonical morphism.

Findings

Full description of Rees algebra equations for ideals of almost-linear type.

Propagation of injectivity of the canonical morphism component.

Examples illustrating scope and applications of the results.

Abstract

In this dissertation, we tackle the problem of describing the equations of the Rees algebra of I for I =(J,y), with J being of linear type. Throughout, such ideals are referred to as ideals of almost-linear type. In Theorem A, we give a full description of the equations of Rees algebras of ideals of the form I = (J,y), with J satisfying an homological vanishing condition. Theorem A permits us to recover and extend well-known results about families of ideals of almost-linear type due to W.V. Vasconcelos, S. Huckaba, N.V. Trung, W. Heinzer and M.-K. Kim, among others. In Theorem B, we prove that the injectivity of a single component of the canonical morphism from the symmetric algebra of I to the Rees algebra of I, propagates downwards, provided I is of almost-linear type. In particular, this result gives a partial answer to a question posed by A.B. Tchernev. Packs of examples are introduced in each section, illustrating the scope and applications of each of the results presented. The author also gives a collection of computations and examples which motivate ongoing and future research.