Additive actions on complete toric surfaces

Abstract

By an additive action on an algebraic variety X we mean a regular effective action G n a × X → X with an open orbit of the commutative unipotent group G n a . In this paper, we give a classification of additive actions on complete toric surfaces.

On uniqueness of additive actions on complete toric varieties

“…Therefore, a maximal unipotent subgroup of the group Aut(X) has dimension 2, but there is no additive action on the variety X by Lemma 1. Now let us recall the main result of [15] and explain the connection between this result and Theorem 4. Definition 7.…”

“…In [15], all additive actions on a complete toric surface were classified. It turns out that there are no more than two non-isomorphic additive actions on a complete toric surface, see Section 7 of this work for more details.…”

“…After presenting some preliminaries on toric varieties and Cox ring (Section 2) and G aactions and Demazure roots (Section 3), we describe the results of [6] (Section 4). In Section 5, we recall some facts on Demazure roots of a toric variety admitting an additive action from [15]. In Section 6, we prove the main result of the paper.…”

Equivariant completions of affine spaces

“…Also we present two results of Dzhunusov. The first one is a classification of additive actions on complete toric surfaces [34], and the second one is a criterion of uniqueness of an additive action on a complete toric variety [35].…”

Additive actions on toric projective hypersurfaces

“…There are several results on additive actions on projective hypersurfaces [4,6,17], flag varieties [1,10,13,14], singular del Pezzo surfaces [9], weighted projective planes [2] and general toric varieties [5,11,12].…”