Quotients of Divisorial Toric Varieties

TL;DR

This paper studies subtorus actions on divisorial toric varieties, providing a characterization for when such actions admit categorical quotients, and introduces a universal reduction technique using support maps.

Contribution

It generalizes previous results for quasiprojective cases by characterizing quotients in divisorial toric varieties and develops a universal reduction method via support maps.

Findings

Characterization of subtorus actions admitting categorical quotients

Universal reduction of toric varieties to divisorial ones

Decomposition of invariant maps into toric and non-toric parts

Abstract

We consider subtorus actions on divisorial toric varieties. Here divisoriality means that the variety has many Cartier divisors like quasiprojective and smooth ones. We characterize when a subtorus action on such a toric variety admits a categorical quotient in the category of divisorial varieties. Our result generalizes previous statements for the quasiprojective case. An important tool for the proof is a universal reduction of an arbitrary toric variety to a divisorial one. This is done in terms of support maps, a notion generalizing support functions on a polytopal fan. A further essential step is the decomposition of a given subtorus invariant regular map to a divisorial variety into an invariant toric part followed by a non-toric part.