Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces

Abstract

AbstractLet X be a reduced nonsingular quasiprojective scheme over such that the set of real rational points X() is dense in X and compact. Then X() is a real algebraic variety. Denote by (X(), ) the group of homology classes represented by Zariski closed k-dimensional subvarieties of X(). In this note we show that (X(), ) is a proper subgroup of H1(X(), ) for a nonorientable hyperelliptic surface X. We also determine all possible groups (X(), ) for a real ruled surface X in connection with the previously known description of all possible topological configurations of X.