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On the enumerative geometry of real algebraic curves having many real branches. (English) Zbl 1019.14021
Given a real plane curve of degree \(d\) and genus \(g\geq 1\), having at least \(g\) real banches (connected components of the normalization), the author gives a formula for the number of real plane curves of degree \(d-1\), which cross each real branch of the given curve at one point and with a prescribed multiplicity. A similar statement is proven for real space curves. The idea is close to that suggested in the previous author’s work [Ill. J. Math. 46, 145-153 (2002; Zbl 1007.14011)]. The enumeration of curves is reduced to the count of specific subgroups in the real part of \(\text{Pic}(C)\). The results are illustrated by a number of examples.
MSC:
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H50 Plane and space curves
14P25 Topology of real algebraic varieties
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