Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the toric subvarity corresponding to $\sigma$. This corrsponds to the map \begin{eqnarray*} \pi \colon X_{\Sigma'} \to X_{\Sigma} \end{eqnarray*} where $\Sigma' = \Sigma \cap \{\tau\}$, $\tau$ coming from the barycentic subdivison of $\sigma_k$.

Let $D_{\Sigma}$ is a toric divisor on $X_{\Sigma}$, and consider the divisor \begin{eqnarray} D_{\Sigma'} = \pi^{\#}(D_{\Sigma}) + qE_{\tau} \end{eqnarray} on $X_{\Sigma'}$ where $q \in \mathbb{Q}$ is a rational number and $E_{\tau}$ is the exceptional divisor of $\pi$.

I want to compute the top intersection number $(D_{\Sigma'})^n$.

If $k = n$, then we get \begin{eqnarray} (D_{\Sigma'})^n &=& (\pi^{\#}(D_{\Sigma}) + qE_{\tau} )^n \\ &=& (\pi^{\#}(D_{\Sigma}))^n + (qE_{\tau} )^n \\&=& (D_{\Sigma})^n + q^n (E_{\tau})^n \end{eqnarray} Does anyone have any idea how to treat the general case? Thank you very much!

  • $\begingroup$ In other words, I want to compute the intersection numbers $E_{\tau}^{n-i}(\pi^{\#}(D_{\Sigma}))^i$ for $i = 1, \cdots, n-1$. These should depend on $k$. But only on $k$? $\endgroup$ – cata Apr 6 '15 at 14:07

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