Rigidity and cardinality of moduli space in Real Algebraic Geometry

Let R be a real closed field and let X be an affine algebraic variety over R. We say that X is universally map rigid (UMR for short) if, for each irreducible affine algebraic variety Y over R, the set of nonconstant rational maps from Y to X is finite. A bijective map φ : X → X from an affine algebraic variety over R to X is called weak change of the algebraic structure of X if it is regular and φ −1 is a Nash map preserving nonsingular points. We prove that, when dim(X) ≥1, there exists a set { φ t : Xt → X}t ∈ R of weak changes of the algebraic structure of X such that each X t is UMR and, for each t, s ∈ R with t ≠ s, Xt and Xs are birationally nonisomorphic. As an immediate consequence, we solve the problem about the cardinality of the moduli space of birationally nonisomorphic affine real algebraic structures on a topological space, on an affine real Nash manifold and, when R is the field of real numbers, on a smooth manifold. The answer to this problem was already known in the case of compact smooth manifolds.