Variations of geometric invariant quotients for pairs, a computational approach

TL;DR

This paper investigates geometric invariant theory (GIT) compactifications of pairs consisting of a hypersurface and a hyperplane, providing a comprehensive framework to understand stability and orbit structures, with applications to Fano and Calabi-Yau hypersurfaces.

Contribution

It characterizes all polarizations leading to different GIT quotients and identifies a finite set of one-parameter subgroups for stability analysis.

Findings

Characterization of all polarizations for GIT quotients

Finite set of subgroups for stability determination

Descriptions of maximal and minimal orbits in the GIT setting

Abstract

We study the GIT compactifications of pairs formed by a hypersurface and a hyperplane. We provide a general setting to characterize all polarizations which give rise to different GIT quotients. Furthermore, we describe a finite set of one-parameter subgroups sufficient to determine the stability of any GIT quotient. We characterize all maximal orbits of non stable and strictly semistable pairs, as well as minimal closed orbits of strictly semistable pairs. Our construction gives natural compactifications of the space of log smooth pairs for Fano and Calabi-Yau hypersurfaces.