TITLE: Birational morphism that is not successive blow-down along smooth centers?
QUESTION [6 upvotes]: 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.)

REPLY [11 votes]: 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.