Geometry and algebra of real forms of complex curves

Original · EN

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.