Geometry and algebra of real forms of complex curves

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.