Elementary structure of morphism space between real algebraic varieties

The paper deals with the first systematic study of the spaces of regular and ratinal maps between arbitrary algebraic varieties over a real closed field R. We find conditions under which these spaces are reduced to the space of Zariski locally constant maps, we investigate the finiteness properties of the subspaces of dominating regular and rational maps and, when R is the field of real numbers, we study the topology of the space of regular maps. Our results show that the mentioned map spaces are usually" very small. The realization of such a general study has been possible thanks to the introduction of two new classes of real algebraic invariants: the curve genera and the toric genera."