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Split out a lemma

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aisejohan committed Jun 12, 2018
1 parent f158b52 commit e11fefbc41ef8ba2d55c58cc32a5582bebd0b299
Showing with 59 additions and 59 deletions.
  1. +6 −56 chow.tex
  2. +53 −3 divisors.tex
@@ -10466,63 +10466,13 @@ \subsection{Blowing up lemmas}
\begin{proof}
Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent ideal sheaf
of denominators of $s$. Namely, we declare a local section
$f$ of $\mathcal{O}_X$ to be a local section of $\mathcal{I}$
if and only if $fs$ is a local section of $\mathcal{L}$.
On any affine open $U = \Spec(A)$
of $X$ write $\mathcal{L}|_U = \widetilde{L}$ for some invertible
$A$-module $L$. Then $A$ is a Noetherian domain with fraction field
$K = R(X)$ and we may think of $s|_U$ as an element of
$L \otimes_A K$ (see
Divisors, Lemma \ref{divisors-lemma-locally-Noetherian-K}).
Let $I = \{x \in A \mid xs \in L\}$. Then we see that
$\mathcal{I}|_U = \widetilde{I}$ (details omitted) and hence
$\mathcal{I}$ is quasi-coherent.
\medskip\noindent
Consider the closed subscheme $Z \subset X$ defined by $\mathcal{I}$.
It is clear that $U = X \setminus Z$. This suggests we should blow
up $Z$. Let
$$
\pi : X' =
\underline{\text{Proj}}_X
\left(\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n\right)
\longrightarrow
X
$$
be the blowing up of $X$ along $Z$. The quasi-coherent
sheaf of $\mathcal{O}_X$-algebras
$\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n$
is generated in degree $1$ over $\mathcal{O}_X$.
Moreover, the degree $1$ part is a coherent $\mathcal{O}_X$-module,
in particular of finite type. Hence we see that $\pi$
is projective and $\mathcal{O}_{X'}(1)$ is relatively very ample.
\medskip\noindent
of denominators of $s$, see Divisors, Definition
\ref{divisors-definition-regular-meromorphic-ideal-denominators}.
By Divisors, Lemma \ref{divisors-lemma-blowing-up-denominators}
we get (2), (3), and (4).
By Divisors, Lemma \ref{divisors-lemma-blow-up-integral-scheme}
we have $X'$ is integral. By
Divisors, Lemma \ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}
there exists an effective Cartier divisor $E \subset X'$ such that
$\pi^{-1}\mathcal{I} \cdot \mathcal{O}_{X'} = \mathcal{I}_E$.
Also, by the same lemma we see that $\pi^{-1}(U) \cong U$.
\medskip\noindent
Denote $s'$ the pullback of the meromorphic section $s$ to a meromorphic
section of $\mathcal{L}' = \pi^*\mathcal{L}$ over $X'$.
It follows from the fact that $\mathcal{I}s \subset \mathcal{L}$
that $\mathcal{I}_Es' \subset \mathcal{L}'$. In other words,
$s'$ gives rise to an $\mathcal{O}_{X'}$-linear map
$\mathcal{I}_E \to \mathcal{L}'$, or in other words
a section $t \in \mathcal{L}' \otimes \mathcal{O}_{X'}(E)$.
By Divisors, Lemma \ref{divisors-lemma-characterize-OD} we obtain a unique
effective Cartier divisor $D \subset X'$ such that
$\mathcal{L}' \otimes \mathcal{O}_{X'}(E) \cong \mathcal{O}_{X'}(D)$
with $t$ corresponding to $1_D$. Reversing this procedure
we conclude that
$\mathcal{L}' = \mathcal{O}_{X'}(-E) \cong \mathcal{O}_{X'}(D)$
with $s'$ corresponding to $1_D \otimes 1_E^{-1}$ as in (4).
\medskip\noindent
we get (1). By Divisors, Lemma \ref{divisors-lemma-blowing-up-projective}
the morphism $\pi$ is projective.
We still have to prove (5).
By Lemma \ref{lemma-equal-c1-as-cycles} we have
$$
@@ -8105,9 +8105,12 @@ \section{Blowing up}
\begin{lemma}
\label{lemma-blow-up-pullback-effective-Cartier}
Let $X$ be a scheme. Let $b : X' \to X$ be a blowup of $X$ in a closed
subscheme. For any effective Cartier divisor $D$ on $X$ the pullback
$b^{-1}D$ is defined (see Definition
\ref{definition-pullback-effective-Cartier-divisor}).
subscheme. The pullback $b^{-1}D$ is defined
for all effective Cartier divisors $D \subset X$
and pullbacks of meromorphic functions are defined for $b$
(Definitions
\ref{definition-pullback-effective-Cartier-divisor} and
\ref{definition-pullback-meromorphic-sections}).
\end{lemma}
\begin{proof}
@@ -8799,7 +8802,54 @@ \section{Admissible blowups}
open subschemes $X'_1$ and $X'_2$.
\end{proof}
\begin{lemma}
\label{lemma-blowing-up-denominators}
Let $X$ be a locally Noetherian scheme.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
Let $s$ be a regular meromorphic section of $\mathcal{L}$.
Let $U \subset X$ be the maximal open subscheme such that
$s$ corresponds to a section of $\mathcal{L}$ over $U$.
The blowup $b : X' \to X$ in the ideal of denominators
of $s$ is $U$-admissible. There exists an effective Cartier divisor
$D \subset X'$ and an isomorphism
$$
b^*\mathcal{L} = \mathcal{O}_{X'}(D - E),
$$
where $E \subset X'$ is the exceptional divisor such that the
meromorphic section $b^*s$ corresponds, via the isomorphism,
to the meromorphic section $1_D \otimes (1_E)^{-1}$.
\end{lemma}
\begin{proof}
From the definition of the ideal of denominators in
Definition
\ref{definition-regular-meromorphic-ideal-denominators}
we immediately see that $b$ is a $U$-admissible blowup.
For the notation $1_{D'}$, $1_E$, and $\mathcal{O}_{X'}(D - E)$
please see Definition
\ref{definition-invertible-sheaf-effective-Cartier-divisor}.
Finally, note that $b^*s$ is defined by
Lemma \ref{lemma-blow-up-pullback-effective-Cartier}.
Thus the statement of the lemma makes sense.
We can reinterpret the final assertion as saying
that $b^*s$ is a global regular section of
$b^*\mathcal{L}(E)$ whose zero scheme is $D$.
This uniquely defines $D$ hence
to prove the lemma we may work affine locally on $X$ and $X'$.
Assume $X = \Spec(A)$ is affine and
$\mathcal{L} = \mathcal{O}_X$. Shrinking further we may assume
$s = a/b$ with $a, b \in A$ nonzerodivisors.
Then the ideal of denominators of $s$ corresponds
to the ideal $I = \{x \in A \mid xa \in bA\}$.
Recall that $X'$ is covered by spectra of affine blowup
algebras $A' = A[\frac{I}{x}]$ with $x \in I$
(Lemma \ref{lemma-blowing-up-affine}).
In $A'$ we have $b = x (b/x)$ as $b \in I$
and $E$ is cut out by $x$.
Thus if we let $D' \cap \Spec(A')$ be the effective
Cartier divisor cut out by the nonzerodivisor $ab/x$ of $A'$, then
the lemma holds over the open $\Spec(A') \subset X'$ as desired.
\end{proof}

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