A D-modules approach on the equations of the Rees algebra

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 $\alpha :\mathrm{\text{ Sym}}(I)\to \mathrm{\text{ Rees}}(I)$ 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 $\mathcal{𝒦}$.