Geometry of multigraded rings and embeddings of toric varieties

TL;DR

This paper introduces a new method for embedding projective varieties into toric varieties using multigraded rings, preserving key birational properties and generalizing existing constructions for Mori Dream Spaces.

Contribution

It develops a novel approach leveraging homogeneous spectra of multigraded rings to construct toric embeddings that extend known methods for Mori Dream Spaces.

Findings

Constructed toric embeddings that retain birational geometry.

Generalized the embedding technique to a broader class of varieties.

Extended the framework of Mori Dream Spaces to new contexts.

Abstract

We use homogeneous spectra of multigraded rings to construct toric embeddings of a large family of projective varieties which preserve some of the birational geometry of the underlying variety, generalizing the well-known construction associated to Mori Dream Spaces.