Specializing Weak Factorization to Birational morphism

Question

The Weak Factorization Theorem tells us that birational map of varieties over field of perfect characteristic which has resolution of singularities can be factored into blow-ups and blow-downs. My question is what happens when we restrict ourself to birational morphisms instead of birational maps? Can we then assume that it is just a sequence of blow-ups, i.e. that a birational morphism $X\to Y$ can be factored as $$X_0\to X_1\to \ldots \to X_n$$ where $X_{i}=\text{Bl}_{D_{i+1}}(X_{i+1})$ where $D_{i+1}\subset X_{i+1}$ is a closed subset.

Answer 1

It is true that every birational morphism is the blow-up of the sheaf of ideals on the range, see theorem 7.17 in Hartshorne.

I don't think that every birational morphism is the blow-uo if a closed subset. Consider (one of the) the small resolution of the $3$-fold ODP with exceptional divisor $\mathbb{CP}^1$. If one blows up the singular point downstairs then gets the full resolution with exceptional divisor $\mathbb{P}^1 \times \mathbb{P}^1$, so this morphism is not the blow-up of a closed subset.

Answer 2

This is true for surfaces for example. See [1] for general result. But in dimension >2 I am quite sure there will be an example of a birational morphism $f:X\to Y$ which cannot be factored as a composition of blow-up.

[1] Stacks Project : https://stacks.math.columbia.edu/tag/0C5R