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?


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    $\begingroup$ 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". $\endgroup$ – Nick Feb 13 '19 at 13:36
  • $\begingroup$ 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 $\endgroup$ – Жека Feb 13 '19 at 15:53
  • $\begingroup$ @Nick : one could have intermediate example as well, for example the blow-up of $\Bbb C^2$ is toric and not affine or projective. $\endgroup$ – Nicolas Hemelsoet Feb 13 '19 at 22:28

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.

  • $\begingroup$ I think this answer could do with more detail: I agree that the wall between the cones corresponds to a curve, but how do you know that curve is projective? $\endgroup$ – Asal Beag Dubh Feb 14 '19 at 10:43
  • $\begingroup$ @AsalBeagDubh : Dear Asal, thanks for your comment, I edited the answer and I hope it's better now. $\endgroup$ – Nicolas Hemelsoet Feb 14 '19 at 11:48

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