Almost proper GIT-stacks and discriminant avoidance (Q612870)

It is proven that the classifying stack of a geometrically reductive group scheme over a field is very close to being proper. Using this result authors show that the proof of the existence of rational sections of \(d\)-dimensional isotrivial family of polarized varieties \((X \to S,\) \({\mathcal N} \in \mathrm{Pic}(X))\) can be reduced to the case when the discriminant of the family is empty. The fibre \((V, \mathcal L)\) of the family \(X\to S\) is assumed to have an automorphism group whose component of unity has a reductive reduction.