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Additive actions on toric varieties. (English) Zbl 1375.14155
The main object of the paper under review is an irreducible algebraic variety $$X$$ of dimension $$n$$ (defined over an algebraically closed field of characteristic zero) admitting an action of the commutative unipotent group $$\mathbb G^n_a$$ (for short, an additive action). The study of such actions goes back to a paper by B. Hassett and Yu. Tschinkel [Internat. Math. Res. Notices 22, 1211–1230 (1999; Zbl. 0966.14033)] who established, in a particular case where $$X$$ is the projective space, a correspondence between such actions and local $$(n + 1)$$-dimensional commutative associative algebras with unit and classified such actions for small $$n$$. Since then many other cases have been settled, and the authors present a brief survey along with a short description of arithmetic motivation. Their focus is on the case where $$X$$ is an arbitrary toric variety. In this case, the first step is to consider actions normalized by the torus $$T$$ acting on $$X$$ (for short, normalized actions). It is known that normalized $$\mathbb G_a$$-actions are in one-to-one correspondence with several objects of different nature: combinatorial (so-called Demazure roots of the fan $$\Sigma$$ defining $$X$$), group-theoretic (certain $$\mathbb G_a$$-subgroups of automorphisms of the Cox ring of $$X$$), or ring-theoretic (homogeneous locally nilpotent derivations of the Cox ring).
The main results of the paper are the following ones. First, Theorem 3.4 asserts that additive actions on a toric variety $$X$$ normalized by the acting torus $$T$$ are in bijection with so-called complete collections of Demazure roots of the fan $$\Sigma$$. Second, they prove in Theorem 3.6 that such a normalized action is unique up to conjugation by an automorphism of $$X$$. Third, they consider arbitrary additive actions, show that $$X$$ admits a normalized action whenever it admits an arbitrary one, and provide a simple geometric criterion for such a situation. Furthermore, for the case where $$X$$ is projective, they provide an elegant criterion for the existence of an additive action in terms of the corresponding polytope. They finish by considering explicit examples and open questions.

MSC:
 14L30 Group actions on varieties or schemes (quotients) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 13N15 Derivations and commutative rings 14J50 Automorphisms of surfaces and higher-dimensional varieties 14M17 Homogeneous spaces and generalizations
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