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Article overview
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Construction of universal Thom-Whitney-a stratifications, their functoriality and Sard-type Theorem for singular varieties | D.Grigoriev
; P.Milman
; | Date: |
9 Nov 2008 | Abstract: | {f Construction.} For a dominating polynomial mapping {$F: K^n o K^l$}
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 {f 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 $Scap S’$ being open and dense in both $S
$ and $ S’ $.
{f 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 {f 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 o Z $ meaning that the sum of the absolute values of the $l imes
l$ 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). | Source: | arXiv, 0811.1373 | Services: | Forum | Review | PDF | Favorites |
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