IUMJ

Title: On elimination of variables in the study of singularities in positive characteristic

Authors: Angelica Benito and Orlando Eugenio Villamayor Uriburu

Issue: Volume 64 (2015), Issue 2, 357-410

Abstract:

The objective of this paper is to discuss some invariants of singularities of algebraic schemes over fields of positive characteristic, and to show their usefulness for the simplification of singularities. We focus here on invariants that arise in an inductive manner, namely, by successive elimination of variables. When applied to a hypersurface singularity, they lead to a refinement of the notion of multiplicity. The Weierstrass Preparation Theorem allows us to express the equation defining the hypersurface as a polynomial equation; in this case, the new invariants can be defined, in some way, from the coefficients of this polynomial, and hence this theorem enables us to define invariants in one variable fewer. In this paper, we present a generalized form of the Weierstrass Preparation Theorem that enables us to eliminate several variables at once, and to define invariants in this more general setting. This leads to the definition of inductive invariants that refine the multiplicity in the hypersurface case. In addition, they provide a refinement of the Hilbert-Samuel stratification for the non-hypersurface case. Finally, the paper includes some applications of these invariants to the open problem of resolution of singularities.