Monoidal transforms and invariants of singularities in positive characteristic

TL;DR

This paper explores monoidal transforms and invariants to address the resolution of singularities in positive characteristic, proposing a strong monomial case enabling combinatorial resolution methods similar to characteristic zero.

Contribution

It introduces invariants that define a strong monomial case in positive characteristic, facilitating a combinatorial resolution approach analogous to characteristic zero.

Findings

Defined invariants for the strong monomial case

Proved resolution can be achieved combinatorially in this case

Extended methods from characteristic zero to positive characteristic

Abstract

The problem of resolution of singularities in positive characteristic can be reformulated as follows: Fix a hypersurface $X$, embedded in a smooth scheme, with points of multiplicity at most $n$. Let an $n$-sequence of transformations of $X$ be a finite composition of monoidal transformations with centers included in the $n$-fold points of $X$, and of its successive strict transforms. The open problem (in positive characteristic) is to prove that there is an $n$-sequence such that the final strict transform of $X$ has no points of multiplicity $n$ (no $n$-fold points). In characteristic zero, such an $n$-sequence is defined in two steps: the first consisting in the transformation of $X$ to a hypersurface with $n$-fold points in the so called monomial case. The second step consists in the elimination of these $n$-fold points (in the monomial case), which is achieved by a simple combinatorial procedure for choices of centers. The invariants treated in this work allow us to define a notion of strong monomial case which parallels that of monomial case in characteristic zero: If a hypersurface is within the strong monomial case we prove that a resolution can be achieved in a combinatorial manner.