Multiplicity of the special fiber of blowups

Abstract

Let (R, m) be a Noetherian local ring and let I be an m-primary ideal. In this paper we give sharp bounds on the multiplicity of the special fiber ring F of I in terms of other well-known invariants of I. A special attention is then paid in studying when equality holds in these bounds, with a particular interest in the unmixedness or, better, the Cohen-Macaulayness of F.