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As well-known in toric geometry, we can define a toric divisor $D_i$ with respect to every face $F_i$ of the polytope. And it is well know that $-K_M=\sum D_i$. My question is that if we denote $D_i$ to be the toric divisor with respect to the face $F_i$, then is $-D_i$ also a toric divisor with respect to $F_i$?

The question comes from a represent of $-K_M$ while $M$ is the blow-up of $\mathbb{C}P^2$ at one point. We know that $-K_M=3H-E$, where $H$ is the hyperplane line bundle and $E$ is the exceptional divisor. So three of $D_i$s are $H$ and the other is $-E$. Are there some mistake?

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    $\begingroup$ If you blow up one of the three torus-fixed points $p$ in $\mathbb{P}^2$, then the resulting surface has four torus-invariant prime Weil divisors. One is the strict transform of the unique torus-invariant line in $\mathbb{P}^2$ that does not contain $p$; this divisor has class $H$. Two are the strict transforms of the torus-invariant lines that contain $p$; these each have class $H-E$. The last is $E$. $\endgroup$
    – Jason Starr
    Commented Jun 12, 2016 at 13:30
  • $\begingroup$ @JasonStarr Thanks for your response. So $-E$ is also torus invariant. $\endgroup$
    – Daniel
    Commented Jun 12, 2016 at 13:58
  • $\begingroup$ The divisor class $-E$ is not represented by a torus-invariant prime Weil divisor. The divisor class $E$ is represented by a torus-invariant prime Weil divisor. $\endgroup$
    – Jason Starr
    Commented Jun 12, 2016 at 14:23
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    $\begingroup$ The basic confusion is that "divisor" is an overused word in algebraic geometry. Sometimes it means a subscheme pure of codimension $1$, sometimes it means an element of the class group. There is a $T$-invariant subscheme $TV(F_i)$ associated to $F_i$, and it has a class maybe we should call $D_i$. Then $-D_i$ means something, but $-TV(F_i)$ means nothing. Your question seems to be: Is the sign of the divisor class associated to a facet ill-defined? The answer is no; the sign is well-defined. $\endgroup$
    – Allen Knutson
    Commented Jun 12, 2016 at 15:24
  • $\begingroup$ @JasonStarr Would you like to provide me some reference where I can find the description of divisors with respect to the faces of one-point blow-up in $\mathbb{P}^2$. Thank you so much! $\endgroup$
    – Daniel
    Commented Jun 13, 2016 at 1:45

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