Projective embeddings of homogeneous spaces with small boundary

TL;DR

This paper investigates special projective embeddings of homogeneous spaces where the boundary is minimal, providing criteria for their existence, finiteness results, and a combinatorial description of their geometric properties.

Contribution

It introduces new criteria for the existence of small boundary embeddings, proves finiteness of their isomorphism classes, and describes their properties using Cox's construction and bunched rings.

Findings

Criteria for existence of small boundary embeddings

Finiteness of isomorphism classes for a given homogeneous space

Description of embeddings' properties via combinatorial methods

Abstract

We study open equivariant projective embeddings of homogeneous spaces such that the complement of the open orbit does not contain divisors. Criterions of existence of such an embedding are considered and finiteness of isomorphism classes of embeddings for a given homogeneous space is proved. Any embedding with small boundary is realized as a GIT-quotient associated with a linearization of the trivial line bundle on the space of the canonical embedding. The generalized Cox's construction and the theory of bunched rings allow us to describe basic geometric properties of embeddings with small boundary in combinatorial terms.