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Blowing-up construction of maximal smoothings of real plane curve singularities. (English) Zbl 0861.14022
Broglia, Fabrizio (ed.) et al., Real analytic and algebraic geometry. Proceedings of the international conference, Trento, Italy, September 21-25, 1992. Berlin: Walter de Gruyter. 169-188 (1995).
Let \(f(x,y)=0\) be the germ at \(O\in \mathbb{R}^2\) of a (reduced) real analytic singularity. We identify it with its analytic representative in a suitable neighborhood of \(O\) and we look at local real deformations \(f_\varepsilon\) of it into a smooth curve (in other words, at local level sets of real morsifications of the singularity); we call such curves smoothings of \(f\). The first natural question, which goes back to V. I. Arnol’d [in: Singularities, Summer Inst., Arcata 1981, Proc. Symp. Pure Math. 40, Part I, 57-69 (1983; Zbl 0519.58019)] is: What topological types of smoothings are possible for a given singularity? We address the problem of construction of smoothings of real plane curve singularities with the possible maximum of connected components. The technique of construction is by using blowing-ups, and lower bounds on the maximal number of connected components are given in the general case (i.e., in the case of a non irreducible curve). The case of a curve with branches having distinct tangents is especially emphasized.
For the entire collection see [Zbl 0812.00016].
14H20 Singularities of curves, local rings
14P05 Real algebraic sets
14N10 Enumerative problems (combinatorial problems) in algebraic geometry