We introduce a notion of generic real algebraic variety and we study the space of morphisms into these varieties. Let be a real algebraic variety. We say that is generic if there exist a finite family of irreducible real algebraic curves with genus and a biregular embedding of into the product variety . A bijective map from a real algebraic variety to is called weak change of the algebraic structure of 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 and for integer , there exists an algebraic family of weak changes of the algebraic structure of such that , is the identity map on and, for each , is generic. Let and be nonsingular irreducible real algebraic varieties. Regard the set of regular maps from to as a subspace of the corresponding set of Nash maps, equipped with the compact-open topology. We prove that, if is generic, then is closed and nowhere dense in , and has a semi-algebraic structure. Moreover, the set of dominating regular maps from to is finite. A version of the preceding results in which and can be singular is given also.
@article{ASNSP_2006_5_5_3_419_0, author = {Ghiloni, Riccardo}, title = {On the space of morphisms into generic real algebraic varieties}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {419--438}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 5}, number = {3}, year = {2006}, mrnumber = {2274786}, zbl = {1170.14309}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2006_5_5_3_419_0/} }
TY - JOUR AU - Ghiloni, Riccardo TI - On the space of morphisms into generic real algebraic varieties JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2006 SP - 419 EP - 438 VL - 5 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2006_5_5_3_419_0/ LA - en ID - ASNSP_2006_5_5_3_419_0 ER -
%0 Journal Article %A Ghiloni, Riccardo %T On the space of morphisms into generic real algebraic varieties %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2006 %P 419-438 %V 5 %N 3 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2006_5_5_3_419_0/ %G en %F ASNSP_2006_5_5_3_419_0
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://archive.numdam.org/item/ASNSP_2006_5_5_3_419_0/
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