The Stacks project

Exercise 111.27.10. Blowing up: Part III. Consider $A$, $I$ and $U$, $Z$ as in the definition of strict transform. Let $Z = V({\mathfrak p})$ for some prime ideal ${\mathfrak p}$. Let $\bar A = A/{\mathfrak p}$ and let $\bar I$ be the image of $I$ in $\bar A$.

1. Show that there exists a surjective ring map $R: = Bl_ I(A) \to \bar R: = Bl_{\bar I}(\bar A)$.

2. Show that the ring map above induces a bijective map from $\text{Proj}(\bar R)$ onto the strict transform $Z'$ of $Z$. (This is not so easy. Hint: Use 5(b) above.)

3. Conclude that the strict transform $Z' = V_{+}(P)$ where $P \subset R$ is the homogeneous ideal defined by $P_ d = I^ d \cap {\mathfrak p}$.

4. Suppose that $Z_1 = V({\mathfrak p})$ and $Z_2 = V({\mathfrak q})$ are irreducible closed subsets defined by prime ideals such that $Z_1 \not\subset Z_2$, and $Z_2 \not\subset Z_1$. Show that blowing up the ideal $I = {\mathfrak p} + {\mathfrak q}$ separates the strict transforms of $Z_1$ and $Z_2$, i.e., $Z_1' \cap Z_2' = \emptyset$. (Hint: Consider the homogeneous ideal $P$ and $Q$ from part (c) and consider $V(P + Q)$.)