A complete toric surface admitting an additive action has exactly one such action if its fan is wide, and two non-isomorphic actions otherwise.
Commutative algebraic monoid structures on affine spaces
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abstract
We study commutative associative polynomial operations $\mathbb{A}^n\times\mathbb{A}^n\to\mathbb{A}^n$ with unit on the affine space $\mathbb{A}^n$ over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.
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Additive actions on complete toric surfaces
A complete toric surface admitting an additive action has exactly one such action if its fan is wide, and two non-isomorphic actions otherwise.