Equivariant completions of affine spaces

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.

68 pages; a subsection on Euler-symmetric varieties is added

References in corpus (9)

• Additive actions on toric varieties
• The automorphism group of a rigid affine variety
• Classifying local Artinian Gorenstein algebras
• Fano threefolds as equivariant compactifications of the vector group
• Commutative algebraic monoid structures on affine spaces
• Equivariant compactifications of a nilpotent group by $G/P$
• Smooth non-projective equivariant completions of affine spaces
• Limit points and additive group actions
• Euler-symmetric complete intersection in projective space