Decomposition of a morphism with positive dimensional fibers

It is well known that any birational morphism between projective varieties is a sequence of blow ups. Suppose now that I have a morphism $$f:X \to Y$$ with positive dimensional fibers, that is a projective bundle over an open subset of $$Y$$. We can even assume $$Y$$ smooth, even if I don't think it is necessary. Is it still true that $$X$$ is the blow-up of a projective bundle on $$Y$$?

• No that is not true. Consider the projective Abelian cone associated to a reflexive sheaf on $Y$ that is not locally free. Aug 16, 2020 at 8:41
• For an explicit example, consider a general hypersurface of bidegree $(1,1)$ in $\mathbb{P}^2\times \mathbb{P}^3$ with its projection to the second factor. Aug 16, 2020 at 10:16
• @IMeasy I'm guessing "projective bundle" = "projective space bundle". Aug 16, 2020 at 12:11
• @Donu : yes, of course that’s what I meant Aug 17, 2020 at 21:09

I am posting my comment as an answer. This already fails for relative dimension $$1$$ when the base scheme has dimension $$n$$ at least $$3$$.
Let $$k$$ be a field. Let $$n\geq 3$$ be an integer. Denote $$\text{Proj}\ k[x_0,x_1,x_2, \dots,x_n]$$ by $$\mathbb{P}^n_k$$. Denote $$\text{Proj}\ k[y_0,y_1,y_2]$$ by $$\mathbb{P}^2_k$$. Denote by $$X$$ the hypersurface in $$\mathbb{P}^2_k\times_{\text{Spec}\ k}\mathbb{P}^n_k$$ with bihomogeneous defining equation, $$f=x_0y_0 + x_1y_1 + x_2y_2.$$ The projection from $$X$$ to $$\mathbb{P}^2_k$$ is a Zariski-locally-trivial projective space bundle of relative dimension $$n-1$$. In particular, $$X$$ is a smooth $$k$$-scheme. By the Grothendieck-Lefschetz Theorem on Picard groups from SGA 2, the restriction homomorphism of Picard groups is an isomorphism, $$\text{res}:\text{Pic}(\mathbb{P}^2_k\times_{\text{Spec}\ k}\mathbb{P}^n_k) \xrightarrow{\cong} \text{Pic}(X).$$ Of course the first Picard group is $$\mathbb{Z}\times \mathbb{Z}$$. Moreover the nef cone in the first Picard group is $$\mathbb{Z}_{\geq 0}\times \mathbb{Z}_{\geq 0}$$. One way to see this is to consider restriction of ample invertible sheaves to linear rational curves ("lines") in fibers of each projection. Since the closed subscheme $$X$$ contains such lines as well, it follows that the restriction isomorphism also induces an isomorphism of nef cones.
In particular, the ample cone of $$X$$ equals $$\mathbb{Z}_{>0}\times \mathbb{Z}_{>0}$$, so that the non-ample nef divisors are those in the "boundary" of the nef cone, i.e., $$\{(0,0)\}\sqcup \left(\mathbb{Z}_{>0}\times \{0\}\right) \sqcup\left( \{0\}\times \mathbb{Z}_{>0}\right).$$ The invertible sheaf in the first component of this partition of the boundary is just the structure sheaf, and the associated contraction of $$X$$ is the constant $$k$$-morphism to $$\text{Spec}\ k$$. The second component gives the projection to $$\mathbb{P}^2_k$$, and the third component gives the projection to $$\mathbb{P}^n_k$$. Since none of these contractions is birational, it follows that $$X$$ is not a blowing up of some projective scheme, except as a "blowing up" that is an isomorphism.
Thus, the projection morphism from $$X$$ to $$\mathbb{P}^n$$ does not factor through a nontrivial blowing up. The restriction of this projection is flat over the closed subscheme $$\text{Zero}(x_0,x_1,x_2)$$, and the restriction is flat over the open complement of this closed subscheme. However, the fiber dimension over the closed subscheme is $$2$$, whereas the fiber dimension over the open subscheme is $$1$$. Thus, this projection is not a projective space bundle, although it is a projective space bundle of relative dimension $$2$$, resp. of relative dimension $$1$$, when restricted over the closed subscheme, resp. over the open subscheme.