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

Abstract

Let I⊂R=𝔽[x1,x2] be a height two ideal minimally generated by three homogeneous polynomials of the same degree d, where 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 𝒦 be the kernel of the canonical map α: Sym(I)→ Rees(I) from the symmetric algebra of I onto the Rees algebra of I. We prove that 𝒦 can be described as the solution set of a system of differential equations, that the whole bigraded structure of 𝒦 is characterized by the integral roots of certain b-functions, and that certain de Rham cohomology groups can give partial information about 𝒦.