# Geometry and algebra of real forms of complex curves

## Authors

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

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## Abstract.

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.