Congruences for real algebraic curves on an ellipsoid

Abstract

The problem of arrangement of a real algebraic curve on a real algebraic surface is related to the 16th Hilbert problem. We prove in this paper new restrictions on arrangement of nonsingular real algebraic curves on an ellipsoid. These restrictions are analogues of Gudkov-Rokhlin, Gudkov-Krakhnov-Kharlamov, Kharlamov-Marin congruences for plane curves (see e.g. \cite{V} or \cite{V1}). To prove our results we follow Marin approach \cite{Marin} that is a study of the quotient space of a surface under the complex conjugation. Note that the Rokhlin approach \cite{R} that is a study of the 2-sheeted covering of the surface branched along the curve can not be directly applied for a proof of Theorem \ref{Rokhlin} since the homology class of a curve of Theorem \ref{Rokhlin} can not be divided by 2 hence such a covering space does not exist.