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1. CMB 2008 (vol 51 pp. 125)
Explicit Real Cubic Surfaces The topological classification of smooth real
cubic surfaces is
recalled and compared to the classification in terms of
the number of real lines and of real tritangent planes,
as obtained
by L.~Schl\"afli in 1858.
Using this, explicit examples of
surfaces of every possible type are given.
Categories:14J25, 14J80, 14P25, 14Q10 |
2. CMB 2001 (vol 44 pp. 257)
Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces 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 $X$.
Categories:14P05, 14P25 |
3. CMB 1999 (vol 42 pp. 445)
Smooth Maps and Real Algebraic Morphisms Let $X$ be a compact nonsingular real algebraic variety and let $Y$
be either the blowup of $\mathbb{P}^n(\mathbb{R})$ along a linear
subspace or a nonsingular hypersurface of $\mathbb{P}^m(\mathbb{R})
\times \mathbb{P}^n(\mathbb{R})$ of bidegree $(1,1)$. It is proved
that a $\mathcal{C}^\infty$ map $f \colon X \rightarrow Y$ can be
approximated by regular maps if and only if $f^* \bigl( H^1(Y,
\mathbb{Z}/2) \bigr) \subseteq H^1_{\alg} (X,\mathbb{Z}/2)$, where
$H^1_{\alg} (X,\mathbb{Z}/2)$ is the subgroup of $H^1 (X,
\mathbb{Z}/2)$ generated by the cohomology classes of algebraic
hypersurfaces in $X$. This follows from another result on maps
into generalized flag varieties.
Categories:14P05, 14P25 |
4. CMB 1997 (vol 40 pp. 456)
Approximation of smooth maps by real algebraic morphisms Let $\Bbb G_{p,q}(\Bbb F)$ be the Grassmann space of all
$q$-dimensional $\Bbb F$-vector subspaces of $\Bbb F^{p}$, where $\Bbb F$
stands for $\Bbb R$, $\Bbb C$ or $\Bbb H$ (the quaternions). Here
$\Bbb G_{p,q}(\Bbb F)$ is regarded as a real algebraic variety. The paper
investigates which ${\cal C}^\infty$ maps from a nonsingular real algebraic
variety $X$ into $\Bbb G_{p,q}(\Bbb F)$ can be approximated, in the
${\cal C}^\infty$ compact-open topology, by real algebraic morphisms.
Categories:14P05, 14P25 |