On the space of morphisms into generic real algebraic varieties
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 3, pp. 419-438.

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 ${\left\{{D}_{i}\right\}}_{i=1}^{n}$ of irreducible real algebraic curves with genus $\ge 2$ and a biregular embedding of $Z$ into the product variety ${\prod }_{i=1}^{n}{D}_{i}$. A bijective map $\varphi \phantom{\rule{-0.166667em}{0ex}}:\phantom{\rule{-0.166667em}{0ex}}{\stackrel{˜}{Z}}^{\phantom{1}}\to Z$ from a real algebraic variety $\stackrel{˜}{Z}$ 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 ${\left\{{\varphi }_{t}:{\stackrel{˜}{Z}}_{t}\to Z\right\}}_{t\in {ℝ}^{k}}$ of weak changes of the algebraic structure of $Z$ such that ${\stackrel{˜}{Z}}_{0}=Z$, ${\varphi }_{0}$ is the identity map on $Z$ and, for each $t\in {ℝ}^{k}\setminus \left\{0\right\}$, ${\stackrel{˜}{Z}}_{t}$ is generic. Let $X$ and $Y$ be nonsingular irreducible real algebraic varieties. Regard the set $ℛ\left(X,Y\right)$ of regular maps from $X$ to $Y$ as a subspace of the corresponding set $𝒩\left(X,Y\right)$ of Nash maps, equipped with the ${C}^{\infty }$ compact-open topology. We prove that, if $Y$ is generic, then $ℛ\left(X,Y\right)$ is closed and nowhere dense in $𝒩\left(X,Y\right)$, 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.

Classification: 14P05,  14P20
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Ghiloni, Riccardo. On the space of morphisms into generic real algebraic varieties. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 3, pp. 419-438. http://archive.numdam.org/item/ASNSP_2006_5_5_3_419_0/

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