Geodesic
Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces
Canadian mathematical bulletin, Tome 44 (2001) no. 3, pp. 257-265

Voir la notice de l'article provenant de la source Cambridge University Press

Let $X$ be a reduced nonsingular quasiprojective scheme over $\mathbb{R}$ such that the set of real rational points $X\left( \mathbb{R} \right)$ is dense in $X$ and compact. Then $X\left( \mathbb{R} \right)$ is a real algebraic variety. Denote by $H_{k}^{a\lg }\left( X\left( \mathbb{R} \right),\,\mathbb{Z}/2 \right)$ the group of homology classes represented by Zariski closed $k$ -dimensional subvarieties of $X\left( \mathbb{R} \right)$ . In this note we show that $H_{1}^{a\lg }\left( X\left( \mathbb{R} \right),\,\mathbb{Z}/2 \right)$ is a proper subgroup of ${{H}_{1}}\left( X\left( \mathbb{R} \right),\,\mathbb{Z}/2 \right)$ for a nonorientable hyperelliptic surface $X$ . We also determine all possible groups $H_{1}^{a\lg }\left( X\left( \mathbb{R} \right),\,\mathbb{Z}/2 \right)$ for a real ruled surface $X$ in connection with the previously known description of all possible topological configurations of $X$ .
DOI : 10.4153/CMB-2001-025-4
Mots-clés : 14P05, 14P25 
Abánades, Miguel A. Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces. Canadian mathematical bulletin, Tome 44 (2001) no. 3, pp. 257-265. doi: 10.4153/CMB-2001-025-4
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     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2001-025-4/}
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