Multi-Harnack smoothings of real plane branches

The 16th problem of Hilbert addresses the determination and the understanding of the possible topological types of smooth real algebraic curves of a given degree in the projective planeRP . This paper is concerned with a local version of this problem: given a germ (C, 0) of real algebraic plane curve singularity, determine the possible topological types of the smoothings of C. A smoothing of C is a real analytic family Ct ⊂ B, for t ∈ [0, 1], such that C0 = C and Ct is non singular and transversal to the boundary of a Milnor ball B of the singularity (C, 0) for 0 < t ≪ 1. In this case the real part RCt of Ct consists of finitely many ovals and non closed components in the Milnor ball. In the algebraic case it was shown by Harnack that a real projective curve of degree d has at most 1 2 (d − 1)(d − 2) + 1 connected components. A curve with this number of components is called a M-curve. In the local case there is a similar bound, depending on the number of real branches of the singularity (see Section 5.1), which arises from the application of the classical topological theory of Smith. A smoothing which reaches this bound on the number of connected components is called a M-smoothing. It should be noticed that in the local case M-smoothings do not always exists (see [K-O-S]). One relevant open problem in the theory is to determine the actual maximal number of components of a smoothing of (C, 0), for C running in a suitable form of equisingularity class refining the classical notion of Zariski of equisingularity class in the complex world (see [K-R-S]). Quite recently Mikhalkin has proved a beautiful topological rigidity property of those M-curves in RP 2 which are embedded in maximal position with respect to the coordinate lines (see [M]). His result, which holds more generally, for those M-curves in projective toric surfaces which are cyclically in maximal position with respect to the toric coordinate lines, is proved by analyzing the topological properties of the associated amoebas. The amoeba of a curve C is the image of the points (x, y) ∈ (C) in the curve by the map Log : (C) → R, given by (x, y) 7→ (log |x|, log |y|). Conceptually, the amoebas are intermediate objects which lay in between classical algebraic curves and tropical curves. See [F-P-T], [G-K-Z], [M], [M-R], [P-R] and [I1] for more on this notion and its applications. In this paper we study smoothings of a real plane branch singularity (C, 0), i.e., the germ (C, 0) is analytically irreducible in (C, 0) and admits a real Newton-Puiseux parametrization. Risler proved that any such germ (C, 0) admits a M-smoothing with the maximal number ovals, namely 1 2 μ(C)0, where μ denotes the Milnor number. The technique used, called nowadays the blow-up method, is a generalization of the classical Harnack construction of M-curves by small perturbations, using the components of the exceptional divisor as a basis of rank one (see [R2], [K-R] and [K-R-S]). One of our motivations was to study to which extent Mikhalkin’s result holds for smoothings of singular points of real plane curves, particularly for Harnack smoothings, those M-smoothings which are in maximal position with respect to two coordinates lines through the singular point.