Geometry and algebra of real forms of complex curves
S. M. Natanzon
Abstract
Let Y be a complex algebraic curve and let [Y]=X1,...,Xn be the set of all real algebraic curves Xi with complexification Xi(C)=Y, such that the real points Xi(R) divide Xi(C). We find all such families [Y]. According to Harnak theorem a number |Xi| of connected components of Xi(R) satifies by the inequality |Xi|<= g+1, where g is the genus of Y. We prove that SUM |Xi| <= 2g-(n-9) 2n-3-2 <= 2g+30 and these estimates are exact.
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