Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces

Canadian mathematical bulletin

关键词: positive solution;    conformal scalar curvature equation;    growth estimate;   

DOI  :  10.4153/CMB-2001-025-4

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $X$ be a reduced nonsingular quasiprojective scheme over ${mathbbR}$ such that the set of real rational points $X({mathbb R})$ is densein $X$ and compact. Then $X({mathbb R})$ is a real algebraic variety.Denote by $H_k^{alg}(X({mathbb R}), {mathbb Z}/2)$ the group ofhomology classes represented by Zariski closed $k$-dimensionalsubvarieties 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 hyperellipticsurface $X$. 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 previouslyknown description of all possible topological configurations of $X$.

【 授权许可】

Unknown