Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces

http://dx.doi.org/10.4153/CMB-2001-025-4
Canad. Math. Bull. 44(2001), 257-265

  Published:2001-09-01
 Printed: Sep 2001

Let $X$ be a reduced nonsingular quasiprojective scheme over ${\mathbb R}$ such that the set of real rational points $X({\mathbb R})$ is dense in $X$ and compact. Then $X({\mathbb R})$ is a real algebraic variety. Denote by $H_k^{\alg}(X({\mathbb R}), {\mathbb Z}/2)$ the group of homology classes represented by Zariski closed $k$-dimensional subvarieties of $X({\mathbb R})$. In this note we show that $H_1^{\alg} (X({\mathbb R}), {\mathbb Z}/2)$ is a proper subgroup of $H_1(X({\mathbb R}), {\mathbb Z}/2)$ for a nonorientable hyperelliptic surface $X$. We also determine all possible groups $H_1^{\alg}(X({\mathbb R}), {\mathbb Z}/2)$ for a real ruled surface $X$ in connection with the previously known description of all possible topological configurations of $X$.