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Some examples of real algebraic and real pseudoholomorphic curves. (English) Zbl 1256.14066
Itenberg, Ilia (ed.) et al., Perspectives in analysis, geometry, and topology. On the occasion of the 60th birthday of Oleg Viro. Based on the Marcus Wallenberg symposium on perspectives in analysis, geometry, and topology, Stockholm, Sweden, May 19–25, 2008. Basel: Birkhäuser (ISBN 978-0-8176-8276-7/hbk; 978-0-8176-8277-4/ebook). Progress in Mathematics 296, 355-387 (2012).
The most interesting classes of real algebraic and pseudoholomorphic curves in the plane (or in another real surface) are constituted by curves with extremal topological characteristics, for example, \(M\)-curves whose total Betti number of the real point set coincides with that of the complex point set. The author constructs several new examples and series of examples of curves having maximal possible for a given degree value of either the number of nonempty ovals, or of the number of ovals of the maximal depth, or of the Milnor number \(n\) of a singularity \(A_n\) on a curve. One more series of examples of real algebraic curves of degree \(4d+1\) meets the conditions of the Orevkov-Viro congruence (such curves with \(d\geq3\) have not been known before). Finally, algebraic \(M\)-curves of degree \(9\) with a single exterior oval are classified up to isotopy.
For the entire collection see [Zbl 1230.00045].

14P25 Topology of real algebraic varieties
14H20 Singularities of curves, local rings
14H50 Plane and space curves
32Q65 Pseudoholomorphic curves
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