Real Algebraic and Analytic Geometry Preprint Server

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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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