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Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces

Published online by Cambridge University Press:  20 November 2018

Miguel A. Abánades
Miguel A. Abánades*
Affiliation:
Department of Mathematics and Statistics University of New Mexico Albuquerque, New Mexico 87131-1141 U.S.A., e-mail: abanades@math.unm.edu
*
Current address: IT Informática de Sistemas CES Felipe II C/ Capitan 39 Aranjuez 28300, Madrid Spain
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Abstract

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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 $X$.

Keywords

14P0514P25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 3 , 01 September 2001 , pp. 257 - 265
Copyright
Copyright © Canadian Mathematical Society 2001

Footnotes

The author was partially supported by NSF Grant DMS-9503138.

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