Theorem 19.3.19B00AB5301F7A

Proof. By Hironaka's Chow lemma theorem 9.10.8, we may assume that $\pi $ is a modification.

We begin by setting $W_0=Z$. We will construct $W_ i$ inductively for each $i$. Assume that for $0\leq i<n$ a smooth partial flag $W_0\supseteq \cdots \supseteq W_ i$ has been constructed on a modification $\pi _ i:Z_ i\rightarrow Z$ so that $\pi \circ \pi _ i$ restricts to bimeromorphic morphisms $W_ j\rightarrow Y_ j$ for each $j=0,\ldots ,i$.

By Zariski's main theorem, $W_ i\rightarrow Y_ i$ is an isomorphism outside a codimension $2$ subset of $Y_ i$. We let $W_{i+1}$ be the strict transform of $Y_{i+1}$ in $W_ i$. The problem is that $W_{i+1}$ is not necessarily smooth.

We will further modify $Z_ i$ and lift $W_1,\ldots , W_{i+1}$ in order to make the flag smooth. Take the embedded resolution of $(W_{j},W_{i+1})$, say $W_ j'\rightarrow W_ j$ for each $j=0,\ldots ,i$.

We have canonical embeddings $W_{i}'\hookrightarrow W_{i-1}'\hookrightarrow \cdots \hookrightarrow W_0'$ making the following diagram commutative:

Let $W_{i+1}'$ be the strict transform of $W_{i+1}$ in $W_ i'$. It suffices to define $\pi _{i+1}$ as the morphism $W_0'\rightarrow Z_ i\rightarrow Z$ and replace $W_0\supseteq \cdots \supseteq W_{i+1}$ by $W_0'\supseteq \cdots \supseteq W_{i+1}'$. $\square$