Elimination with Applications to Singularities in Positive Characteristic

Abstract

We present applications of elimination theory to the study of singularities over arbitrary fields. A partial extension of a function, defining resolution of singularities over fields of characteristic zero, is discussed here in positive characteristic.

Singularities in positive characteristic, stratification and simplification of the singular locus

“…Thus, Main Theorem 10.1 provides a characteristic free form of induction; and as a consequence, a simplification of singularities over arbitrary characteristic (in the spirit of the reduction to monomial case) can be obtained (this is shown in [32]). In other words, the so-called "reduction to the monomial case" is possible in positive characteristic.…”

“…Therefore, this part of the inductive argument, used in characteristic zero, can be extended to positive characteristic. The iteration of blow-ups at the maximum strata of this function, leads to a simplification of the singularities in some lower dimension (see [32]). …”

“…To do so, first choose a positive integer n, and an element f ∈ I n of order n at O V (d) ,x , and then produce a monic polynomial f (Z) ∈ I n as in (32) in a suitable étale neighborhood of x. Then, it can be checked that, up to integral closure, we may assume that G is as in (37) This elimination algebra depends on the projection β d,d−1 but by construction it does not depend on the election of f once the projection is fixed, and it satisfies the following conditions:…”

Techniques for the study of singularities with applications to resolution of 2-dimensional schemes

“…This last result suggests that embedded resolution of singularities should hold, at least for 3-dimensional schemes over perfect fields. More results concerning resolution of singularities in positive characteristic can be found in [1,3,7,16,25,26,[28][29][30]38,39,42,43,48].…”

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“…There is a novel approach to resolution by Villamayor and his collaborators Benito, Bravo and Encinas [Vi2,Vi3,BV,EV3]. It is based on projections instead of restrictions for the descent in dimension.…”