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].

For the entire collection see [Zbl 0812.00016].