Topology of real algebraic curves: A Felix Klein question (Q1594969)

Symmetric surfaces, i.e., compact connected orientable surfaces equipped with an involution that reverses the orientation, arise as the sets of complex points of smooth irreducible projective real algebraic curves. Klein's problem is to determine which symmetric surfaces can be derived from real plane curves. Topologically, symmetric surfaces are classified by three invariants: the genus \(g\), the number, \(r\), of connected components of the real locus, and the separating number \(a\) (which is defined to be 0 or 1 according as the remaining surface is disconnected or connected after removing the real locus). In the case of symmetric surfaces in general and also for symmetric surfaces arising from plane curves there are well-known restrictions on the possible values of these invariants, e.g., bounds for the number \(r\) in terms of the genus. It is shown that all values of the invariants that are not forbidden by the known restrictions do, in fact, occur, i.e., there are no more independent restrictions for the existence of symmetric surfaces.