Let $X_0 \subset X$ a toroidal embedding without self intersections and denote by $\overline{\Sigma}$ its corresponding (weakly embedded) extended conical simplicial complex. Let $D$ be a divisor on $X$ supported away from $X_0$. Then $D$ corresponds to a continuous integral function on \begin{eqnarray*} \psi \colon |\Sigma| \to \mathbb{R} \end{eqnarray*} which restricted to each cone $\sigma \in \Sigma$ is linear. I want to compute the self intersection number \begin{eqnarray*} D^n \in A_0(X) \simeq \mathbb{Z} \end{eqnarray*} where $n = \dim(X)$.

I would like to do this using tropical intersection theory on weakly embedded extended cone complexes http://arxiv.org/abs/1510.04604. For this, I have to find a divisor \begin{eqnarray*}F \sim D \end{eqnarray*} linearly equivalent to $D$ which intersects the open set $X_0$ non trivially and has transversal intersections with all boundary strata.

How can I do this?

Is there another way to compute this intersection number combinatorially on the cone complex?


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