Bounding the degrees of the defining equations of Rees rings for certain determinantal and Pfaffian ideals

TL;DR

This paper studies the algebraic structure of Rees rings associated with determinantal and Pfaffian ideals, providing bounds on their defining equations and conditions for certain properties based on ideal heights.

Contribution

It characterizes the $G_s$ condition for these ideals and derives bounds on the degrees of generators of their Rees rings through specialization techniques.

Findings

Characterization of $G_s$ condition in terms of minors and Pfaffians heights.

Bounds on generation and concentration degrees of Rees rings.

Explicit degree bounds for specific classes of ideals.

Abstract

We consider ideals of minors of a matrix, ideals of minors of a symmetric matrix, and ideals of Pfaffians of an alternating matrix. Assuming these ideals are of generic height, we characterize the condition $G_{s}$ for these ideals in terms of the heights of other ideals of minors or Pfaffians of the same matrix. We additionally obtain bounds on the generation and concentration degrees of the Rees rings of a subclass of such ideals via specialization of the Rees rings in the generic case. We do this by proving, given sufficient height conditions on ideals of minors or Pfaffians of the matrix, the specialization of a resolution of the Rees ring in the generic case is an approximate resolution of the Rees ring in question. We end the paper by giving some explicit generation and concentration degree bounds.