Equivariant completions of affine space

In this talk we survey recent results on open embeddings of the complex affine space $\mathbb{A}^n$ into a complete algebraic variety $X$ such that the action of the vector group $G$ on $\mathbb{A}^n$ by translations extends to an action of $G$ on $X$. We begin with Hassett-Tschinkel correspondence describing equivariant embeddings of $\mathbb{A}^n$ into projective spaces and give its generalization for embeddings into projective hypersurfaces. We prove that non-degenerate projective hypersurfaces admitting such an embedding are in bijection with Gorenstein local algebras. Moreover, such an embedding into a projective hypersurface is unique if and only if the hypersurface is non-degenerate.Further we deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types. Supported by the Russian Science Foundation grant 23-21-00472