Diffeomorphisms, isotopies, and braid monodromy factorizations of plane cuspidal curvesDifféomorphismes, isotopies et décompositions monodromiques de courbes planes cuspidales

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Abstract

In this paper we prove that there is an infinite sequence of pairs of plane cuspidal curves, Cm,1 and Cm,2, of degree deg(Cm,1)=deg(Cm,2)→∞, such that the pairs (CP2,Cm,1) and (CP2,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 conjugate.

Résumé

Nous construisons deux suites infinies de courbes planes cuspidales, {Cm,1} et {Cm,2} de degré degCm,1=degCm,2→∞, pour lesquelles les paires (CP2,Cm,1) et (CP2,Cm,2) sont difféomorphes, tandis que Cm,1 et Cm,2 ont des décompositions monodromiques non équivalentes. Ces courbes ne sont pas dans la même classe de déformations équisingulières, ni même isotopes. Dans nos exemples, Cm,2 est déduite de Cm,1 par la conjugaison complexe.

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  • On braid monodromy factorizations

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Partially supported by INTAS-OPEN-97-2072, NWO-RFBR-047-008-005, and RFBR 99-01-01133. This work was done during the stay of the second author in Strasbourg university.

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