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 D_1,...,D_n of irreducible real algebraic curves with genus ≥ 2 and a biregular embedding of Z into the product variety D_1 x ... x D_n. A bijective map 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 {F_t : Z'_t → Z}_{t in R^k} (where R denotes the field of real numbers) of weak changes of the algebraic structure of Z such that Z'_0 = Z, F_0 is the identity map on Z and, for each t in R^k \ {0}, Z'_t is generic. Let X and Y be nonsingular irreducible real algebraic varieties. Regard the set RM(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 RM(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.