Mathematische Zeitschrift

, Volume 243, Issue 2, pp 391–407

Geometry and algebra of real forms of complex curves


  • S.M. Natanzon
    • Moscow State University Korp. A, Leninske Gory, 11899 Moscow, Russia
Original article

DOI: 10.1007/s00209-002-0480-0

Cite this article as:
Natanzon, S. Math Z (2003) 243: 391. doi:10.1007/s00209-002-0480-0


Let Y be a complex algebraic curve and let \([Y]=\{X_1,...,X_n\}\) be the set of all real algebraic curves \(X_i\) with complexification \(X_i({\Bbb C})=Y\), such that the real points \(X_i({\Bbb R})\) divide \(X_i({\Bbb C})\). We find all such families [Y]. According to Harnak theorem a number \(\vert X_i\vert\) of connected components of \(X_i({\Bbb R})\) satisfies by the inequality \(\vert X_i\vert\leqslant g+1\), where g is the genus of Y. We prove that \(\sum\vert X_i\vert \leqslant 2g-(n-9) 2^{n-3}-2\leqslant 2g+30\) and these estimates are exact.

Mathematics Subject Classification: 14H35, 14E09, 30F50, 14P25

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© Springer-Verlag Berlin Heidelberg 2002