Variation of Stratifications From Toric GIT

TL;DR

This paper investigates how stratifications induced by group actions on toric varieties vary with different ample divisors, identifying conditions and walls that govern these variations.

Contribution

It provides criteria for when different ample divisors induce the same stratification and describes walls in the ample cone that control stratification changes.

Findings

Identified conditions for identical stratifications from different divisors

Formulated two types of walls in the ample cone affecting stratification variation

Showed variation depends on primitive collections and fan relations

Abstract

When a reductive group acts on an algebraic variety, a linearized ample line bundle induces a stratification on the variety where the strata are ordered by the degrees of instability. In this paper, we study variation of stratifications coming from the group actions in the GIT quotient construction for projective toric varieties. Cox showed that each projective toric variety is a GIT quotient of an affine space by a diagonalizable group with respect to linearizations that come from ample divisors on the toric variety. We provide a sufficient conditions for two ample divisors to induce the same stratification and formulate two types of walls in the ample cone that completely capture two kinds of variations. We also prove that the variation is intrinsic to the primitive collections and the relations among ray generators of the fans.