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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.

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    $\begingroup$ All this information is encoded in the fan of a toric variety. $\endgroup$
    – Sasha
    Dec 15 '16 at 13:33
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    $\begingroup$ It's given by the unique torus invariant section of the anticanonical bundle. $\endgroup$ Dec 15 '16 at 14:44
  • $\begingroup$ @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. $\endgroup$
    – THC
    Dec 15 '16 at 16:45
  • $\begingroup$ @THC: Yes, everything is very explicit in terms of the fan. $\endgroup$
    – Sasha
    Dec 15 '16 at 19:50
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Here is a gentle lecture note Minicourse on Toric varieties by D. A. Cox with illustrative examples, hope it helps.

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Invariant toric subvarieties are just orbit closures. In the affine case there is a simple combinatorial description of such subvarieties in terms of quotient fans. Details can be found, for example, in the book by Günter Ewald entitled "Combinatorial convenite and algebraic geometry", GTM 168, Springer

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