Abstract view
Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces


Published:20010901
Printed: Sep 2001
Abstract
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 $X$.