Some combinatorial remarks on normal flatness in analytic spaces

Abstract

In this article we present a combinatorial treatment of normal flatness in analytic spaces, using the idea of equimultiple standard bases. We will prove, using purely combinatorial methods, a characterization theorem for normal flatness. This will lead us to a new proof of a classical theorem on normal flatness, which can be stated by saying that normal flatness at a point along a smooth subspace is equivalent to the Hilbert function being locally constant. Though these topics belong to classical analytic geometry, we believe that this approach is valuable, since it replaces extremely general algebraic theorems by combinatorial objects, obtaining new results and striking the combinatorial nature of the classical (and basic) ideas in the resolution of singularities.