Inherently Improper Parametric Supports for Unirational Varieties

A class of parametric supports in the lattice space Z is found to be inherently improper because any rational parametrization from C to C defined on such a support is improper. For a generic rational parametrization defined on an inherently improper support, we prove that its improper index is the gcd of the normalized volumes of all the simplex sub-supports and give an algorithm to obtain a proper reparametrization. Properties of non-degenerate rational parametrizations defined on an inherently improper parametric support with coefficients from some subfield of C are also considered. Finally, the coordinate and lattice structures of inherently improper parametric supports are analyzed and a reparametrization algorithm of better complexity is given.