On the highest multiplicity locus of algebraic varieties and Rees algebras

Let X be an equidimensional scheme of finite type over a perfect field k. Under these conditions, the multiplicity along points of X defines an upper semi-continuous function, say multX:X→N, which stratifies X into its locally closed level sets. We study this stratification, and the behavior of the multiplicity when blowing up at regular equimultiple centers. We also discuss a natural compatibility of these two concepts when X is replaced with its underlying reduced scheme. The main result in this paper is to show that, given a variety X, there is a well defined Rees algebra over X, naturally attached to maximum value of the multiplicity.