Linking Blow-ups

Question

I would like to know when we can "zig-zag-connect" blow-ups of the same base space, by which I mean the following: Let $X$ be a $k$-scheme and $Z_1,Z_2$ two closed subschemes and $\text{Bl}_{Z_i}X$ the associated blow-ups. Under which conditions does there exist a closed subscheme $D\subset X$ such that we have $X$-morphisms $$\text{Bl}_{Z_1}X\leftarrow\text{Bl}_{D}X \rightarrow \text{Bl}_{Z_2}X$$ The problem is that I don't understand the pullback of $Z_i$ under $\pi_D:\text{Bl}_{D}X \rightarrow X$, or rather when it is an effective Cartier divisor, so that I can leverage the universal property. Since $\pi_D:\text{Bl}_{D}X\backslash E \rightarrow X\backslash D$ is an isomorphism, we have $\pi_D^{-1}(Z_i\backslash D)\cong Z_i\backslash D$, so unless $Z_i$ is a divisor, we probably want $D\supset Z_i$. So essentially the question is when is the total transform of a closed subset of $D$ an effective divisor? Does this for instance work when everything in sight is smooth?

Answer 1

Let $f_1: Bl_{Z_1}X \to X$ and $f_2: Bl_{Z_2}X \to X$. Now we have an isomoprphism $$Bl_{f_1^{-1}(Z_2)}\Big(Bl_{Z_1}X\Big)\simeq Bl_{f_2^{-1}(Z_1)}\Big(Bl_{Z_2}X\Big).$$

For a proof I would simply refer to Eisenbud+Harris's Geometry of Schemes, Lemma IV-41.

Answer 2

If $X = \mathrm{Spec}(A)$ is affine and $Z_i = \mathrm{Spec}(A/\mathfrak a_i)$ for some ideals $\mathfrak a_1,\mathfrak a_2$, then the pullback of $Z_i \to X$ is just $Z_1 \cap Z_2$ (trivially in set-theoretic terms, but these schemes need not be reduced, so $Z_1 \times_X Z_2$ is a more accurate description of that scheme... but then the statement becomes redundant so this was just for intuition), which in terms of ideals corresponds to $\mathrm{Spec}(A/(\mathfrak a_1 + \mathfrak a_2))$ since we have the isomorphism $$ A / \mathfrak a_1 \otimes_A A / \mathfrak a_2 \simeq A / (\mathfrak a_1 + \mathfrak a_2). $$ I'm pretty sure the universal property of the blow-up allows you to produce the required morphisms out of that diagram. I worked locally (in an affine setting) but you can do it globally as well.

You can think of this $\mathrm{Bl}_D(X)$ as just blowing up along the intersection $Z_1 \cap Z_2$ and inserting that "blown-up intersection" inside the individual blown-ups.

This means that if $\mathrm{Bl}_{Z_i}(X) = \mathrm{Proj}(A[\mathfrak a_i t])$, then $\mathrm{Bl}_D(X) = \mathrm{Proj}(A[(\mathfrak a_1 + \mathfrak a_2) t])$ works, and the inclusion maps $A[\mathfrak a_i t] \subseteq A[(\mathfrak a_1 + \mathfrak a_2) t]$ induce the morphisms $\mathrm{Bl}_D(X) \to \mathrm{Bl}_{Z_i}(X)$.

Feel free to turn that into relative projs and use sheaves of ideals, the ideas will be the same. For instance, you'll have $\mathrm{Bl}_{Z_i}(X) = \mathbf{Proj}_X(\mathcal O_X[\mathscr I_i t])$ if the closed subscheme $Z$ is defined by the sheaf of ideals $\mathscr I_i \trianglelefteq \mathcal O_X$, and $D$ will be defined by the sheaf of ideals $\mathscr I_1 + \mathscr I_2$.

Hope that helps,