Parametric Galois extensions

Given a field k and a finite group H, an H-parametric extension over k   is a finite Galois extension of k(T)k(T) of Galois group containing H which is regular over k and has all the Galois extensions of k of group H among its specializations. We are mainly interested in producing non-H-parametric extensions, which relates to classical questions in inverse Galois theory like the Beckmann–Black problem and the existence of one parameter generic polynomials. We develop a general approach started in a preceding paper and provide new non-parametricity criteria and new examples.