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Lipschitz stratification of complex hypersurfaces in codimension 2. (English) Zbl 07714602

Summary: We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study the Zariski equisingular families of surface, not necessarily isolated, singularities in \(\mathbb{C}^3\). We show that a natural stratification of such a family, given by the singular set and the generic family of polar curves, provides a Lipschitz stratification in the sense of Mostowski. In particular such families are bi-Lipschitz trivial, with trivializations obtained by integrating Lipschitz vector fields.

MSC:

14B05 Singularities in algebraic geometry
32Sxx Complex singularities
32B10 Germs of analytic sets, local parametrization
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[1] S /. Then it is enough to extend vj S to a Lipschitz vector field on S [ 0 , since then such an extension defines a Lipschitz vector field on S [ .s/ [ 0 .s/ for every s sufficiently small, with the Lipschitz constant independent of s. Case 2: dist. .s/; 0 .s// dist. 0 .s/;
[2] S /. Then it suffices to extend v from to a Lip-schitz vector field on [ 0 . Note that we may suppose that on both arcs , 0 we have y D O.x/, z D O.x/, that is, they are in the form (32). Indeed, by the Transversality Assumptions the variable z restricted to an arc in X cannot dominate x and y, that is, x D o.z/;
[3] small. This is a change of coordinates in the target of the projection .x; y; z; t / 7 ! .x; y; t / and affects neither the discriminant locus nor Zariski’s equisingularity. To make the proof more precise we will use the constant “ of Definition 4.1 and denote the resulting union of polar wedges and the singular set by P W ” . If both .s/; 0 .s/ belong to P W ” then the claim follows from the first part of the proof (Section 6). In Case 1, given a stratified Lipschitz vector field v onto S we extend it on 0 .
[4] By Proposition 9.1 we may suppose that dist. .s/; C j / & s m j for every j , and there-fore, for b small, say b Ä ”, dist. .s/; C j / dist. .s/;
[5] Ä ”=2. Thus there exists a quasi-wing QW containing 0 and moreover dist. 0 .s/;
[6] S / D dist. b 0 . 0 .s//; b 0 / s l , where l D max ¹max l i ; max r k º and b 0 denotes the discrim-inant of b 0 . Then there is a Lipschitz extension of v to QW by Proposition 9.4. Similarly, in Case 2 we may suppose dist. .s/; C j / dist. 0 .s/; C j / & s m j for every j , since otherwise, by Proposition 9.1, both .s/; 0 .s/ belong to P W ” . Then, choosing b appropriately, we may suppose that dist. b . .s//; b . 0 .s/// dist. .s/; 0 .s// s l :
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