On the space of morphisms into generic real algebraic varieties

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 {Di}i=1 of irreducible real algebraic curves with genus ≥ 2 and a biregular embedding of Z into the product variety Qn i=1 Di. A bijective map φ : e Z −→ Z from a real algebraic variety e 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 {φt : e Zt −→ Z}t∈Rk of weak changes of the algebraic structure of Z such that e Z0 = Z, φ0 is the identity map on Z and, for each t ∈ R \ {0}, e Zt is generic. Let X and Y be nonsingular 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∞ 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. Mathematics Subject Classification: Primary 14P05; Secondary 14P20.