Algebraic homology for real hyperelliptic and real projective ruled surfaces

Let X be a reduced nonsingular quasiprojective scheme over R such that the set of real rational points X(R) is dense in X and compact. Then X(R) is a real algebraic variety. Denote by H-k(alg) (X(R), Z/2) the group of homology classes represented by Zariski closed k-dimensional subvarieties of X(R). In this note we show that H-l(alg) (X(R), Z/2) is a proper subgroup of H-l (X(R), Z/2) for a nonorientable hyperelliptic surface X. We also determine all possible groups H-l(alg) (X(R), Z/2) for a real ruled surface X in connection with the previously known description of all possible topological configurations of X.