On the length of the absolute samuel stratum

Abstract

Let X be a scheme of finite type over the field of complex numbers and let x be the generic point of the center of a permissible monoidal transformation X′ → X. If the generic point y of the projectivization of the strict tangent space T x X is an infinitely near H-fold point to X at x (i.e., the Hilbert-Samuel function of X′ at y “coincides” with the Hilbert-Samuel function of X at x) then the length of the Samuel stratum of X passing through x strictly decreases whenever it is finite.