# Birational morphism that is not successive blow-down along smooth centers?

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.)

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.

• Possibly silly question, but which point has a 2-dimensional fiber? Such $[y_1:\ldots:y_6]\in\mathbb{P}^6$ would have to make the equations identically zero, but at least one term of them will always be non-zero I think. Oct 22, 2021 at 10:36
• @pbelmans: In $\mathbb{P}^6$ we have seven homogeneous coordinates, and so there is a unique point which has $y_1 = \dots = y_6 = 0$. Oct 22, 2021 at 11:04
• Oh yes, I was silly indeed, and I missed $y_0$. What a rookie mistake! Thanks! Oct 22, 2021 at 11:08

Let $$C$$ be a general curve of genus $$g$$ and take $$\pi: Sym^g(C) \to Jac(C)$$.

Then brill-noether says that that fibers of dimension $$r$$ arise from $$h^0=r+1$$ line bundles which form a $$g-h^0 h^1$$ dimensional subspace of $$Jac(C)$$.

We have $$h^0-h^1 = 1$$ for our degree $$g$$ and so generically there is one point in the fiber i.e birational.

But take $$h^0 \cdot h^1 = g, h^0 = 1+h^1$$ so roughly $$h^0, h^1 = \sqrt{g}$$ then there are finitely many points with fiber roughly $$\sqrt{g}$$ dimensional.