Polynomial parametrization and étale exoticity (Q1815240)

This paper relates two properties of constructible sets. The first is the existence of a polynomial parametrization of the set and the second is the étale exoticity of the set. An \(n\)-dimensional complex sphere is the hypersurface in \(\mathbb{C}^{n+1}\) which is defined by the equation \(X^2_1+ \dots+ X^2_{n+1} =1\). It is proved that it has a polynomial parametrization whenever it is even-dimensional. The odd-dimensional case is still open. These polynomial parametrizations were derived via varieties which are generalizations of the étale exotic surface \(S_2\) and of Winkelmann's quadric which is a four-dimensional variety that is embedded in \(\mathbb{C}^5\). The definition of étale exotic surface is generalized to higher dimensions and it is proved that Winkelmann's quadric is a four-dimensional étale exotic variety.