When toric variety is affine?

Question

Let $\Delta$ be a fan and let $X$ be a toric variety associated to $\Delta$. Is there a quick way to tell when $X$ is affine by looking at the fan?

Thanks.

Answer 1

A toric variety is affine if and only if there is a single cone.

Indeed, it's enough to look at the case where $\Sigma$ is the union of two cones $C_1,C_2$ sharing a commun wall. By the orbit-cone correspondence, the wall correspond to a curve $Y \cong \Bbb C^*$. Its closure $\overline{Y}$ is $\Bbb C^* \cup \{p_1\} \cup \{p_2\}$ where $p_i$ is the point corresponding to $C_i$, again using the orbit-cone correspondence which also gives the closure ordering. This shows that $Y$ is complete (even projective since every complete curve is projective) so $X_{\Sigma}$ can't be affine.

Remark : in my previous message I assumed that $Y \cong \Bbb P^1$ but in fact this is wrong as there might be singularities, however it doesn't change the argument.