Equivariant completions of affine spaces

TL;DR

This paper surveys recent advances in extending affine space embeddings into complete algebraic varieties with compatible group actions, covering projective, flag, toric, and Fano varieties.

Contribution

It generalizes the Hassett-Tschinkel correspondence to new classes of embeddings and explores their properties in various geometric contexts.

Findings

Extended the Hassett-Tschinkel correspondence to projective hypersurfaces.

Analyzed equivariant embeddings into flag varieties and degenerations.

Explored embeddings into complete toric and Fano varieties.

Abstract

We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. We begin with Hassett-Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and give its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types.