Open loci of ideals with applications to Birational maps

In this work we show that the loci of ideals in principal class, ideals of grade at least two, and ideals of maximal analytic spread are Zariski open sets in the parameter space. As an application, we show that the set of birational maps of {\it clear polynomial degree} $d$ over an arbitrary projective variety $X$, denoted by $\Bir(X)_{d}$, is a constructible set. This extends a previous result by Blanc and Furter.