# When toric variety is affine?

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.

• By definition, the toric variety of a single cone is affine. I believe there's a theorem that says it's compact (projective) if the fan is a "complete fan". – Nick Feb 13 '19 at 13:36
• Yes, but I think that there is a toric variety associated to a fan with more that one cone and still affine. I would like to understand when this happens. Edit – Жека Feb 13 '19 at 15:53
• @Nick : one could have intermediate example as well, for example the blow-up of $\Bbb C^2$ is toric and not affine or projective. – Nicolas Hemelsoet Feb 13 '19 at 22:28

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.