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.
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Received: 15 November 2001; in final form: 28 April 2002/Published online: 2 December 2002
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Natanzon, S. Geometry and algebra of real forms of complex curves . Math Z 243, 391–407 (2003). https://doi.org/10.1007/s00209-002-0480-0
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DOI: https://doi.org/10.1007/s00209-002-0480-0