On the principle of real moduli flexibility: perfect parametrizations
Volume 111 / 2014
Annales Polonici Mathematici 111 (2014), 245-258
MSC: Primary 14P20; Secondary 14P05, 14P25.
DOI: 10.4064/ap111-3-3
Abstract
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.
