Real Algebraic and Analytic Geometry |

On the space of morphisms into generic real algebraic varieties.

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Submission: 2005, September 1.

*Abstract:
We introduce a notion of generic real algebraic variety and we
study the space of morphisms into these varieties. Let $Z$ be a real
algebraic variety. We say that $Z$ is generic if there exist a finite
family $\{D_i\}_{i=1}^n$ of irreducible real algebraic curves with genus
$\geq 2$ and a biregular embedding of $Z$ into the product variety
$\prod_{i=1}^nD_i$. A bijective map $\varphi:\widetilde{Z} \longrightarrow
Z$ from a real algebraic variety $\widetilde{Z}$ to $Z$ is called weak
change of the algebraic structure of~$Z$ if it is regular and its inverse
is a Nash map. Generic real algebraic varieties are ``generic'' in the
sense specified by the following result: For~each real algebraic variety
$Z$ and for integer $k$, there exists an algebraic family
$\{\varphi_t:\widetilde{Z}_t \longrightarrow Z\}_{t \in \mathbb{R}^k}$ of
weak changes of the algebraic structure of $Z$ such that
$\widetilde{Z}_0=Z$, $\varphi_0$ is the identity map on $Z$ and, for each
$t \in \mathbb{R}^k \setminus \{0\}$, $\widetilde{Z}_t$ is generic. Let
$X$ and $Y$ be nonsingular real algebraic varieties. Regard the set
$\mathcal{R}(X,Y)$ of regular maps from $X$ to $Y$ as a subspace of the
corresponding set $\mathcal{N}(X,Y)$ of Nash maps, equipped with the
$C^{\infty}$~compact--open topo\-logy. We prove that, if $Y$ is generic,
then $\mathcal{R}(X,Y)$~is closed and nowhere dense in $\mathcal{N}(X,Y)$,
and has a semi--algebraic structure. Moreover, the set of dominating
regular maps from $X$ to $Y$ is finite. A version of the preceding results
in which $X$ and $Y$ can be singular is given also.*

Mathematics Subject Classification (2000): 14P05, 14P20.

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