On the principle of real moduli flexibility: perfect parametrizations

Edoardo Ballico¹ ; Riccardo Ghiloni¹

¹ Department of Mathematics University of Trento 38123 Povo-Trento, Italy

Annales Polonici Mathematici, Tome 111 (2014) no. 3, pp. 245-258

Résumé

Let $V$ be a real algebraic manifold of positive dimension. The aim of this paper is to show that, for every integer $b$ (arbitrarily large), there exists a trivial Nash family $\mathscr {V}=\{V_y\}_{y \in R^b}$ of real algebraic manifolds such that $V_0=V$, $\mathscr {V}$ is an algebraic family of real algebraic manifolds over $y \in R^b \setminus \{0\}$ (possibly singular over $y=0$) and $\mathscr {V}$ is perfectly parametrized by $R^b$ in the sense that $V_y$ is birationally nonisomorphic to $V_z$ for every $y,z \in R^b$ with $y \not =z$. A similar result continues to hold if $V$ is a singular real algebraic set.

Détail

DOI : 10.4064/ap111-3-3

Keywords: real algebraic manifold positive dimension paper every integer arbitrarily large there exists trivial nash family mathscr real algebraic manifolds mathscr algebraic family real algebraic manifolds setminus possibly singular mathscr perfectly parametrized sense birationally nonisomorphic every similar result continues singular real algebraic set