Construction of universal Thom-Whitney-a stratifications, their functoriality and Sard-type Theorem for singular varieties

TL;DR

This paper develops a method to construct universal Thom-Whitney-a stratifications for singular varieties using a bundle associated with polynomial maps, and proves a Sard-type theorem for such singular spaces.

Contribution

It introduces a new construction of stratifications based on a bundle G_F and establishes conditions for their universality and existence, along with a Sard-type theorem for singular varieties.

Findings

Universal T-W-a stratifications exist iff fibers of G_F are orthogonal complements to tangent spaces.

The coarsest universal T-W-a stratification is obtained by partitioning Sing(F) via quasistrata.

A Sard-type theorem for singular spaces characterizes noncritical points by uniform noncriticality nearby.

Abstract

{\bf Construction.} For a dominating polynomial mapping {$F: K^n\to K^l$} with an isolated critical value at 0 ($K$ an algebraically closed field of characteristic zero) we construct a closed {\it bundle} $G_F \subset T^{*}K^n $. We restrict $ G_F $ over the critical points $Sing(F)$ of $ F$ in $ F^{-1}(0)$ and partition $Sing(F)$ into {\it 'quasistrata'} of points with the fibers of $G_F$ of constant dimension. It turns out that T-W-a (Thom and Whitney-a) stratifications 'near' $F^{-1}(0)$ exist iff the fibers of bundle $G_F$ are orthogonal to the tangent spaces at the smooth points of the quasistrata (e. g. when $ l=1$). Also, the latter are the orthogonal complements over an irreducible component $ S $ of a quasistratum only if $S $ is {\bf universal} for the class of {T-W-a} stratifications, meaning that for any $\{S_j'\}_j $ in the class, $ \Sing (F) = \cup_j S'_j $, there is a component $S' $ of an $ S_j' $ with $S\cap S'$ being open and dense in both $S $ and $ S' $. {\bf Results.} We prove that T-W-a stratifications with only universal strata exist iff all fibers of $G_F$ are the orthogonal complements to the respective tangent spaces to the quasistrata, and then the partition of $\Sing(F)$ by the latter yields the coarsest {\it universal T-W-a stratification}. The key ingredient is our version of {\bf Sard-type Theorem for singular spaces} in which a singular point is considered to be noncritical iff nonsingular points nearby are 'uniformly noncritical' (e. g. for a dominating map $ F: X \to Z $ meaning that the sum of the absolute values of the $l\times l$ minors of the Jacobian matrix of $ F $, where $ l = \dim (Z) $, not only does not vanish but, moreover, is separated from zero by a positive constant).