Is there an example of a birational morphism of smooth complex projective varieties $f\colon X\to Y$, that cannot be factored into a chain $X\to X\_1\to\cdots\to X\_n\to Y$ of blow-down along smooth centers?


(By weak factorization theorem, we know in general that $f$ can be factorized into a zig-zag of blow-ups and blow-downs along smooth centers.)



Birational morphism that is not successive blow-down along smooth centers?
Let $X \subset \mathbb{P}^2\_{x\_i} \times \mathbb{P}^6\_{y\_j}$ be given by the equations
$$
x\_1y\_1 + x\_2y\_2 + x\_3y\_3 = x\_1y\_4 + x\_2y\_5 + x\_3y\_6 = 0.
$$
It is smooth because its projection to $\mathbb{P}^2$ is a $\mathbb{P}^4$-fibration. This also implies that the rank of the Picard group of $X$ is 2. Now let
$$
f \colon X \to \mathbb{P}^6
$$
be the projection. It is a birational morphism, and if it is a sequence of smooth blowups, it is itself a smooth blowup (because the difference of the Picard ranks is 1). But it is not a smooth blowup, because $f$ has 1-dimensional fibers over a codimension 2 subvariety of $\mathbb{P}^6$ and 2-dimensional fiber over a point.


