GIT stability of divisors in products of projective spaces

Abstract

We study GIT stability of divisors in products of projective spaces. We first construct a finite set of one-parameter subgroups sufficient to determine the stability of the GIT quotient. In addition, we characterise all maximal orbits of not stable and strictly semistable pairs, as well as minimal closed orbits of strictly semistable pairs. This characterisation is applied to classify the GIT quotient of threefolds of bidegree (1,2) and bidegree (4,4) curves in a quadric surface, via singularities, which are in turn used to obtain an explicit description of the K-moduli space of family 2.25 of Fano threefolds, and the K-moduli wall-crossing of log Fano pairs.