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Multi-Harnack smoothings of real plane branches. (English) Zbl 1194.14042

This paper deals with the following problem: given a germ \((C,O)\) of real algebraic plane curve singularity, determine the possible topological types of the smoothings of \(C\). One motivation is to study to which extent G. Mikhalkin’s result [Ann. Math. (2) 151, No. 1, 309–326 (2000; Zbl 1073.14555)] holds for smoothings of singular points of real algebraic plane curves, particularly for Harnack smoothings. A new construction of smoothings of real plane branch \((C,O)\) is developed by using Viro’s patchworking method [see e.g. I. Itenberg and O. Viro, Math. Intell. 18, No. 4, 19–28 (1996; Zbl 0876.14017)] applied to certain Newton non-degenerate curve singularities with several branches.
Then a multi-Harnack smoothing is introduced: it is a \(g\)-parametrical deformation (where \(g\) is the number of characteristic pairs of the branch) which arise as the result of a sequence, beginning at the last step of the resolution, consisting of a suitable Harnack smoothing (in terms Mikhalkin’s definition) followed by the corresponding monomial blow down. One of the main results of the paper is that the multi-Harnack smoothings of a real plane branch \((C,O)\) have a unique topological type which depends only on the complex equisingular class of \((C,O)\).

MSC:

14H20 Singularities of curves, local rings
14P25 Topology of real algebraic varieties
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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