Exercise 111.27.7. Blowing up: part I. In this exercise $R = Bl_ I(A) = A \oplus I \oplus I^2 \oplus \ldots$. Consider the natural map $b : \text{Proj}(R) \to \mathop{\mathrm{Spec}}(A)$. Set $U = \mathop{\mathrm{Spec}}(A) - V(I)$. Show that

$b : b^{-1}(U) \longrightarrow U$

is a homeomorphism. Thus we may think of $U$ as an open subset of $\text{Proj}(R)$. Let $Z \subset \mathop{\mathrm{Spec}}(A)$ be an irreducible closed subscheme with generic point $\xi \in Z$. Assume that $\xi \not\in V(I)$, in other words $Z \not\subset V(I)$, in other words $\xi \in U$, in other words $Z\cap U \not= \emptyset$. We define the strict transform $Z'$ of $Z$ to be the closure of the unique point $\xi '$ lying above $\xi$. Another way to say this is that $Z'$ is the closure in $\text{Proj}(R)$ of the locally closed subset $Z\cap U \subset U \subset \text{Proj}(R)$.

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