Induced additive actions on toric varieties
Alexander Chernov
TL;DR
This work develops a systematic link between induced additive actions on projective toric varieties and $S$-pairs $(A,U)$, where $A$ is a local algebra and $U\subseteq\mathfrak m$ generates $A$. Leveraging the Hassett-Tschinkel correspondence, it proves that linearly normal toric varieties with a normalized additive action correspond to monomial $S$-pairs with $U$ generated by the torus coordinates, and it shows how such pairs yield torus-equivariant, normalized actions (while not guaranteeing normality). The paper then computes explicit $S$-pairs for Hirzebruch surfaces under both normalized and non-normalized actions, and analyzes very ample divisors on $\Sigma_n$, including dimensions of global sections and their toric descriptions. In the $\Sigma_1$ case it conducts a detailed derivation of the relations among the action-generating derivations, using Baker-Campbell-Hausdorff techniques to elucidate the algebraic structure. These results advance the classification and construction of additive actions on toric surfaces and connect geometric embeddings with concrete monomial and differential-algebraic data.
Abstract
An induced additive action on a projective variety X in P^n is a regular action of the group G_a^n on X with an open orbit that can be extended to a regular action on P^n. Such actions are described with pairs (A, U), where A is a local algebra and U is a generating subspace lying in the maximal ideal. Such pairs are called S-pairs. We study additive actions on projective toric varieties and on Hirzebruch surfaces in particular. We prove that for any linearly normal toric variety with an additive action normalized by a torus, the corresponding S-pair consists of a monomial algebra and a subspace generated by variables. Also, we describe S-pairs for normalized and non-normalized induced additive actions on Hirzebruch surfaces.