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Maximal smoothings of real plane curve singular points. (English) Zbl 0988.14025
Turaev, V. (ed.) et al., Topology, ergodic theory, real algebraic geometry. Rokhlin’s memorial. Providence, RI: American Mathematical Society (AMS). Transl., Ser. 2, Am. Math. Soc. 202(50), 167-195 (2001).
Let \((C,0)\) be a real plane singularity. Let \(\mu\) be the Milnor number of \((C,0)\), and let \(r\) be the number of complex branches of \((C,0)\). Let \(v\) be the number of closed connected components, i.e. the number of ovals, of the set of real points of a small smooth real deformation \((\widetilde{C},0)\) of \((C,0)\). The local Harnack inequality states that \(v\leq\frac{1}{2}(\mu-r+1)\), if \((C,0)\) has a real branch, and \(v\leq\frac{1}{2}(\mu-r+3)\), otherwise [J.-J. Risler, Invent. Math. 89, 119-137 (1987; Zbl 0672.14020)]. The deformation \((\widetilde{C},0)\) is called an {\(M\)-smoothing} if equality holds.
\(M\)-smoothings of \((C,0)\) are proven to exist if \(r=1\) (loc. cit.). \(M\)-smoothings of \((C,0)\) do not exist in general [see V. M. Kharlamov, S. Yu. Orevkov and E. I. Shustin, in: The Arnoldfest, Toronto 1997, Fields Inst. Commun. 24, 273-309 (1999; Zbl 0978.14048)].
In the paper under review, the authors extend these results by showing that \((C,0)\) has an \(M\)-smoothing in each of the following cases:
(1) \((C,0)\) is semi-quasi-homogeneous having no peripheral real roots of different signs,
(2) \((C,0)\) is Newton nondegenerate without real branches,
and yet some other cases.
They also show that, for any real plane singularity \((C,0)\), there is a real plane singularity \((C',0)\) that is topologically equivalent to \((C,0)\) over \(\mathbb C\), and that admits an \(M\)-smoothing. The proofs use Viro’s method [O. Ya. Viro, in: Topology conference, Proc., Collect. Rep., Leningrad 1982, 149-197 (1983; Zbl 0605.14021)].
The paper concludes with a sharpened local Harnack inequality, and the corresponding problem of existence of so-called weak \(M\)-smoothings.
For the entire collection see [Zbl 0961.00011].
14P25 Topology of real algebraic varieties
14H20 Singularities of curves, local rings