Diffeomorphisms, Isotopies, and Braid Monodromy Factorizations of Plane Cuspidal Curves
نویسنده
چکیده
In this paper we prove that there is an infinite sequence of pairs of plane cuspidal curves Cm,1 and Cm,2, such that the pairs (CP, Cm,1) and (CP, Cm,2) are diffeomorphic, but Cm,1 and Cm,2 have non-equivalent braid monodromy factorizations. These curves give rise to the negative solutions of ”Dif=Def” and ”Dif=Iso” problems for plane irreducible cuspidal curves. In our examples, Cm,1 and Cm,2 are complex conjugated. Up to our knowledge the following natural questions were open till now: does an existence of a diffeomorphism CP → CP transforming an algebraic curve C1 in an algebraic curve C2 imply the existence of an equisingular deformation (”Dif=Def”) or of a diffeotopie (”Dif=Iso”) between them? We show that the response to the both questions is in the negative. In our examples the curves are cuspidal and irreducible (note that for nodal curves the responses are in the positive, see [1]). Let C ⊂ CP be a plane algebraic curve of degree degC = d. The embedding C ⊂ CP is determined, up to diffeotopy, by the so-called braid monodromy of C (see [4]). The latter depends on a choice of generic homogeneous coordinates in CP and can be seen as a factorization of the full twist ∆2d in the normal semi-group B + d of quasi-positive braids on d strings (∆2d is the so-called Garside element; ∆ 2 d = (σ1 · ... · σd−1) d in standard generators σ1, . . . , σd−1 of B + d ). If the only singularities of C are ordinary cusps and nodes (cuspidal curve), then this factorization can be written as follows
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تاریخ انتشار 2001