Comparing two Proj-like constructions on toric varieties

TL;DR

This paper investigates the relationship between two constructions of toric varieties, establishing a canonical open embedding and criteria for their isomorphism, thereby clarifying their structural connections.

Contribution

It provides a detailed comparison between Perling's toric Proj and Brenner and Schroer's homogeneous spectrum ProjMH, including conditions for their isomorphism.

Findings

Existence of a canonical open embedding between the two constructions.

A criterion for when the two constructions are isomorphic.

Enhanced understanding of the structural relationship between toric Proj and ProjMH.

Abstract

The paper explores the relation between Perling's toric Proj of a multigraded ring A associated with a toric variety (tProj A), and Brenner and Schroer's homogeneous spectrum ProjMH A of the same ring. We show that there is always a canonical open embedding and study a criterion for them to be isomorphic.