Equivariant completions of affine spaces

I. V. Arzhantsev   ; Yu. I. Zaitseva

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 77 (2022) no. 4, pp. 571-650

Résumé

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 the Hassett–Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and present 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. Bibliography: 109 titles.

Détail

Keywords: algebraic variety, additive action, local algebra, projective space, flag variety, grading, locally nilpotent derivation, toric variety, Cox ring, Demazure root.
Mots-clés : affine space, algebraic group, quadric

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