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Hi All,

I'm having some trouble understanding a result about calculating $D.C$ on a toric variety. The proposition I am trying to follow is from Cox, Little, Schenck. Either there is a mistake with the proposition, or I'm doing something wrong in my example - I'm hoping that someone can tell me which one it is. If people have questions about the notation that I will say what they mean, but I think everything is pretty standard.

Let $C=V(\tau)$ be the complete torus invariant curve in $X_\Sigma$ coming from the wall $\tau = \sigma \cap \sigma'$. Let $D$ be a Cartier divisor with Cartier data $m_\sigma$, $m_{\sigma'} \in M$ corresponding to $\sigma, \sigma' \in \Sigma(n)$. Pick $u \in \sigma' \cap N$ that maps to the minimal generator of $\overline{\sigma'}$ in $N(\tau)$. Then $D.C = (m_\sigma - m_{\sigma'})(u)$.

I'm running into problems through the following example. Consider the usual fan for $\mathbb{P}^1 \times \mathbb{P}^1$. Let $\sigma$ be the top right quadrant and let $\sigma'$ be the bottom right quadrant. Let C be the curve corresponding to $\tau = \sigma \cap \sigma'$. Let $D$ be the curve coming from the top vertical ray. We know that $C$ gives us $\mathbb{P}^1 \times p$ and that $D$ is the curve $q \times \mathbb{P}^1$ (maybe I got the order mixed up, but it shouldn't change the calculation). So $D.C=1$. Now let's calculate $D.C$ using the proposition. I found that $D$ has cartier data $m_\sigma =e_1^* -e_2^*$ and $m_\sigma'=e_1^*+e_2^*$, and $u=(1,-1)$ maps to a minimal generator. So then $D.C = -2 e_2^* ((1,-1)) = 2$

Thanks a lot, and apologies if this is just me making a silly blunder.


edit: I figured out my problem - my $m_\sigma'$ is incorrect. I made the same mistake 3 times in a row. Fixing the mistake gives $m_\sigma'=-e_1^*$, which then gives $D.C=1$. Not sure how to close my own thread so this doesn't keep popping up to the front.

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