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

Open loci of ideals with applications to Birational maps