Open loci of ideals with applications to Birational maps

TL;DR

This paper proves that certain loci of ideals are Zariski open and shows that the set of birational maps of fixed degree over a projective variety is constructible, extending previous results.

Contribution

It establishes the Zariski openness of specific ideal loci and extends the understanding of the structure of birational maps of fixed degree.

Findings

Loci of ideals in principal class are Zariski open.

Ideals of grade at least two form Zariski open sets.

Birational maps of fixed degree form a constructible set.

Abstract

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.