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Actually, Nick Proudfoot and I have been talking for years about the irreducible smooth surface constructed from countably many copies of ${\mathbb A}^2$ by gluing $(p,q)$ in the $n$th copy to $(p^2q,p^{-1})$ in the $n+1$th copy. This even has a ${\mathbb G}_m$ action $\lambda \cdot (p,q) = (\lambda p, \lambda^{-1} q)$, a symplectic form $dp \wedge dq$, and a moment map $(p,q) \mapsto pq$ whose zero fiber is an infinite chain of projective lines. This too can be regarded as the toric variety associated to a fan of infinite type in the plane. It appears to be just another way of describing Ekedahl's example.
Actually, Nick Proudfoot and I have been talking for years about the irreducible smooth surface constructed from countably many copies of ${\mathbb A}^2$ by gluing $(p,q)$ in the $n$th copy to $(p^2q,p^{-1})$ in the $n+1$th copy. This even has a ${\mathbb G}_m$ action $\lambda \cdot (p,q) = (\lambda p, \lambda^{-1} q)$, a symplectic form $dp \wedge dq$, and a moment map $(p,q) \mapsto pq$ . The whose zero fiber is an infinite chain of projective lines. This too can be regarded as the toric variety associated to a fan of infinite type in the plane. It appears to be just another way of describing Ekedahl's example.
Actually, Nick Proudfoot and I have been talking for years about the irreducible smooth surface constructed from countably many copies of ${\mathbb A}^2$ by gluing $(p,q)$ in the $n$th copy to $(p^2q,p^{-1})$ in the $n+1$th copy. This even has a ${\mathbb G}_m$ action $\lambda \cdot (p,q) = (\lambda p, \lambda^{-1} q)$, a symplectic form $dp \wedge dq$, and a moment map $(p,q) \mapsto pq$. The zero fiber is an infinite chain of projective lines. This too can be regarded as the toric variety associated to a fan of infinite type in the plane.