9. Resolving singularities of a toroidal embedding in terms of decompositions of polyhedral complexes (09.06.15).

Aim of the talk: Let $U\hookrightarrow X$ be a toroidal embedding with a Cartier divisor $D$ having support equal to $X\setminus U$. Show that the allowable modifications $f: Z\to X$ with $Z$ non-singular and $f^{-1}(D)$ a reduced divisor correspond to certain decompositions of the conical polyhedral complex attached to $U\hookrightarrow X$.

Details: In II, §2 state Theorem 9*- 12*. Prove as much of Theorem 10* and 12* as possible.

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