Canadian Mathematical Society www.cms.math.ca
 location:  Publications → journals → CMB
Abstract view

# Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces

 Read article[PDF: 119KB]
Published:2001-09-01
Printed: Sep 2001
• Miguel A. Abánades
 Format: HTML LaTeX MathJax PDF PostScript

## 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 $X$.
 MSC Classifications: 14P05 - Real algebraic sets [See also 12Dxx, 13P30] 14P25 - Topology of real algebraic varieties

 top of page | contact us | privacy | site map |

© Canadian Mathematical Society, 2017 : https://cms.math.ca/