Let $X$ be $CP^1\times CP^1$ with a point deleted from $\infty$, specifically the point $([1,0],[0,1])$. Is $X$ also a toric variety? What's the fan of $X$?


1 Answer 1


Yes, $X$ is a toric variety. In general, if $Y$ is toric, and $X \subset Y$ is a closed torus-invariant subset, then $Y \setminus X$ is toric too.

As for the fan: a torus-invariant point corresponds to the interior of a top-dimensional cone in the fan. So take the fan of $\mathbf P^1 \times \mathbf P^1$ and throw away the inside of one of the 4 top-dimensional cones.

All of this is described in Fulton's short monograph on toric varieties.


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