A $D$-module approach on the equations of the Rees algebra

TL;DR

This paper employs $D$-module theory to analyze the defining equations of the Rees algebra of a specific ideal in a polynomial ring, linking algebraic structures to differential equations and cohomology.

Contribution

It introduces a novel $D$-module framework to describe the equations of the Rees algebra, connecting algebraic properties with differential equations and cohomological methods.

Findings

The kernel of the canonical map is characterized by solutions to differential equations.

The bigraded structure of the kernel is determined by roots of certain $b$-functions.

De Rham cohomology groups provide partial descriptions of the kernel.

Abstract

Let $I \subset R = \mathbb{F}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{F}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce information about the defining equations of the Rees algebra of $I$. Let $\mathcal{K}$ be the kernel of the canonical map $\alpha: \text{Sym}(I) \rightarrow \text{Rees}(I)$ from the symmetric algebra of $I$ onto the Rees algebra of $I$. We prove that $\mathcal{K}$ can be described as the solution set of a system of differential equations, that the whole bigraded structure of $\mathcal{K}$ is characterized by the integral roots of certain $b$-functions, and that certain de Rham cohomology groups can give partial information about $\mathcal{K}$.