Multiplicity of the special fiber of blowups

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $I$ be an $\mathfrak{m}$-primary ideal. In this paper we give sharp bounds on the multiplicity of the special fiber ring $\mathcal{F}$ 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 $\mathcal{F}$.

14 pages