# The boundary of toric varieties

Let $\mathcal{X}$ be a toric variety, with $T$ a torus embedded as an open set in $\mathcal{X}$ (and where the algebraic action of $T$ extends to $\mathcal{X}$). As I am not a toric specialist at all, I was wondering if there is a simple description of the "boundary," that is, the closed subvariety $C_T := \mathcal{X} \setminus T$. Example: if $\mathcal{X} = \mathrm{Spec}(k[x,y])$ and $T \cong (k^\times)^2$, then $C_T$ consists of two intersecting (affine) lines.

• All this information is encoded in the fan of a toric variety. Dec 15 '16 at 13:33
• It's given by the unique torus invariant section of the anticanonical bundle. Dec 15 '16 at 14:44
• @Sasha: by using (= translating from) the fan, can one give a simple ("concrete") description in the case dimension 2 (I mean, in a similar way as in my question) ? Thanks so much.
– THC
Dec 15 '16 at 16:45
• @THC: Yes, everything is very explicit in terms of the fan. Dec 15 '16 at 19:50