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 {D i } i=1 n of irreducible real algebraic curves with genus 2 and a biregular embedding of Z into the product variety i=1 n D i . A bijective map ϕ:Z ˜ 1 Z from a real algebraic variety 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 {ϕ t :Z ˜ t Z} t k of weak changes of the algebraic structure of Z such that Z ˜ 0 =Z, ϕ 0 is the identity map on Z and, for each t k {0}, Z ˜ t is generic. Let X and Y be nonsingular irreducible real algebraic varieties. Regard the set (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  compact-open topology. We prove that, if Y is generic, then (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.

Classification: 14P05, 14P20
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     title = {On the space of morphisms into generic real algebraic varieties},
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     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
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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://www.numdam.org/item/ASNSP_2006_5_5_3_419_0/

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