Thom Isotopy Theorem for Nonproper Maps and Computation of Sets of Stratified Generalized Critical Values

Sĩ Tiệp Ðinh,Zbigniew Jelonek

doi:10.1007/s00454-019-00087-w

Sĩ Tiệp Ðinh, Zbigniew Jelonek

Open Access

https://doi.org/10.1007/s00454-019-00087-w

Copy DOI

Abstract

Let Xsubset {mathbb {C}}^n be an affine variety and f:Xrightarrow {mathbb {C}}^m be the restriction to X of a polynomial map {mathbb {C}}^nrightarrow {mathbb {C}}^m. We construct an affine Whitney stratification of X. The set K(f) of stratified generalized critical values of f can also be computed. We show that K(f) is a nowhere dense subset of {mathbb {C}}^m which contains the set B(f) of bifurcation values of f by proving a version of the Thom isotopy lemma for nonproper polynomial maps on singular varieties.

Highlights

Ehresmann’s fibration theorem [3] states that a proper smooth surjective submersion f : X → N between smooth manifolds is a locally trivial fibration
B( f ), is called the bifurcation set of f, which is the union of the set K0( f ) of critical values and the set B∞( f ) of bifurcation values at infinity of f
Gh = {g1h, . . . , gsh} is a basis for I h ⊂ k[x0, x1, . . . , xn]. This theorem allows us to compute the set of points at infinity of an affine variety given by the ideal I ; to this aim, it is enough to compute the Gröbner basis {g1, . . . , gs} of the ideal I and to consider the ideal I∞ = (x0, g1h, . . . , gsh)

Summary

Introduction

Ehresmann’s fibration theorem [3] states that a proper smooth surjective submersion f : X → N between smooth manifolds is a locally trivial fibration. For a dominant map f : X → Cm on a smooth complex affine variety X , the computation of K∞( f ), and of the set of generalized critical values, K ( f ) := K0( f ) ∪ K∞( f ), is given in [8,9,10]. To date, to the best of the authors’ knowledge, no connection between B( f ) and the set of stratified generalized critical values of f , defined by K ( f , S) :=. Calculate the set K ( f , S) of stratified generalized critical values of f. The remainder of this manuscript is organized as follows: In Sect. For an algebraic variety X , the singular part and the regular part of X are denoted respectively by sing(X ) and reg(X )

Preliminaries

Construction of Affine Stratifications

Construction of Affine Whitney Stratifications

Thom Isotopy Lemma for Nonproper Maps

Computation of the Sets of Stratified Generalized Critical Values

A Sketch of an Algorithm