On the space of morphisms into generic real algebraic varieties

Ghiloni, Riccardo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 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 $\varphi :{\tilde{Z}}^{\phantom{1}}\to Z$ from a real algebraic variety $\tilde{Z}$ 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 ${\{{\varphi }_t:{\tilde{Z}}_t\to Z\}}_{t\in {ℝ}^k}$ of weak changes of the algebraic structure of $Z$ such that ${\tilde{Z}}_0=Z$, ${\varphi }_0$ is the identity map on $Z$ and, for each $t\in {ℝ}^k\setminus \left\{0\right\}$, ${\tilde{Z}}_t$ is generic. Let $X$ and $Y$ be nonsingular irreducible real algebraic varieties. Regard the set $ℛ(X,Y)$ of regular maps from $X$ to $Y$ as a subspace of the corresponding set $𝒩(X,Y)$ of Nash maps, equipped with the $C^{\infty }$ compact-open topology. We prove that, if $Y$ is generic, then $ℛ(X,Y)$ is closed and nowhere dense in $𝒩(X,Y)$, 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.

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