Finitely supported ⁎-simple complete ideals in a regular local ring

Let I be a finitely supported complete m-primary ideal of a regular local ring (R,m). A theorem of Lipman implies that I has a unique factorization as a ⁎-product of special ⁎-simple complete ideals with possibly negative exponents for some of the factors. The existence of negative exponents occurs if dimR⩾3 because of the existence of finitely supported ⁎-simple ideals that are not special. We consider properties of special ⁎-simple complete ideals such as their Rees valuations and point basis. Let (R,m) be a d-dimensional equicharacteristic regular local ring with m=(x1,…,xd)R. We define monomial quadratic transforms of R and consider transforms and inverse transforms of monomial ideals. For a large class of monomial ideals I that includes complete inverse transforms, we prove that the minimal number of generators of I is completely determined by the order of I. We give necessary and sufficient conditions for the complete inverse transform of a ⁎-product of monomial ideals to be the ⁎-product of the complete inverse transforms of the factors. This yields examples of finitely supported ⁎-simple monomial ideals that are not special. We prove that a finitely supported ⁎-simple monomial ideal with linearly ordered base points is special ⁎-simple.