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.Comment: 19 pg