A D-MODULES APPROACH ON THE EQUATIONS OF THE REES ALGEBRA

Let I subset of R = F[x(1), x(2)] 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 alpha : 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.