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\begin{document}
\title{Resolution of Surfaces}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
This chapter discusses resolution of singularities of surfaces
following Lipman \cite{Lipman} and mostly following the exposition of
Artin in \cite{Artin-Lipman}. The main result
(Theorem \ref{theorem-resolve}) tells us that a Noetherian
$2$-dimensional scheme $Y$ has a resolution of singularities when
it has a finite normalization $Y^\nu \to Y$ with
finitely many singular points $y_i \in Y^\nu$ and for each $i$ the completion
$\mathcal{O}_{Y^\nu, y_i}^\wedge$ is normal.
\medskip\noindent
To be sure, if $Y$ is a $2$-dimensional scheme of finite type over
a quasi-excellent base ring $R$ (for example a field or a
Dedekind domain with fraction field of characteristic $0$
such as $\mathbf{Z}$) then the normalization of $Y$ is finite,
has finitely many singular points, and the completions of the
local rings are normal. See the discussion in
More on Algebra, Sections
\ref{more-algebra-section-singular-locus},
\ref{more-algebra-section-G-ring}, and
\ref{more-algebra-section-excellent}
and
More on Algebra, Lemma \ref{more-algebra-lemma-normal-goes-up}.
Thus such a $Y$ has a resolution of singularities.
\medskip\noindent
A rough outline of the proof is as follows. Let $A$ be a
Noetherian local domain of dimension $2$. The steps of the proof
are as follows
\begin{enumerate}
\item[N] replace $A$ by its normalization,
\item[V] prove Grauert-Riemenschneider,
\item[B] show there is a maximum $g$ of the lengths of
$H^1(X, \mathcal{O}_X)$ over all normal modifications $X \to \Spec(A)$
and reduce to the case $g = 0$,
\item[R] we say $A$ defines a rational singularity if $g = 0$
and in this case after a finite number of
blowups we may assume $A$ is Gorenstein and $g = 0$,
\item[D] we say $A$ defines a rational double point if
$g = 0$ and $A$ is Gorenstein and in this case we
explicitly resolve singularities.
\end{enumerate}
Each of these steps needs assumptions on the ring $A$.
We will discuss each of these in turn.
\medskip\noindent
Ad N: Here we need to assume that $A$ has a finite normalization
(this is not automatic). Throughout most of the chapter we will
assume that our scheme is Nagata if we need to know some normalization
is finite. However, being Nagata is a slightly stronger condition
than is given to us in the statement of the theorem.
A solution to this (slight) problem would have been to use that
our ring $A$ is formally unramified (i.e., its completion
is reduced) and to use Lemma \ref{lemma-formally-unramified}.
However, the way our proof works, it turns out it is easier to
use Lemma \ref{lemma-normalization-completion}
to lift finiteness of the normalization over the
completion to finiteness of the normalization over $A$.
\medskip\noindent
Ad V: This is Proposition \ref{proposition-Grauert-Riemenschneider}
and it roughly states that for a normal modification $f : X \to \Spec(A)$
one has $R^1f_*\omega_X = 0$ where $\omega_X$ is the dualizing module
of $X/A$ (Remark \ref{remark-dualizing-setup}).
In fact, by duality the result is equivalent to a statement
(Lemma \ref{lemma-R1-injective})
about the object $Rf_*\mathcal{O}_X$ in the {\it derived category} $D(A)$.
Having said this, the proof uses the standard fact that
components of the special fibre have positive conormal
sheaves (Lemma \ref{lemma-nontrivial-normal-bundle}).
\medskip\noindent
Ad B: This is in some sense the most subtle part of the proof.
In the end we only need to use the output of this step when $A$
is a complete Noetherian local ring, although the writeup is a
bit more general. The terminology is set in
Definition \ref{definition-reduce-to-rational}.
If $g$ (as defined above) is bounded, then a straightforward
argument shows that we can find a normal modification $X \to \Spec(A)$
such that all singular points of $X$ are rational singularities, see
Lemma \ref{lemma-reduce-to-rational}. We show that given a finite extension
$A \subset B$, then $g$ is bounded for $B$ if it is bounded for $A$
in the following two cases: (1) if the fraction field extension
is separable, see Lemma \ref{lemma-reduce-to-rational} and
(2) if the fraction field extension has degree $p$,
the characteristic is $p$, and $A$ is regular and complete, see
Lemma \ref{lemma-go-up-degree-p}.
\medskip\noindent
Ad R: Here we reduce the case $g = 0$ to the Gorenstein case.
A marvellous fact, which makes everything work, is that the
blowing up of a rational surface singularity is normal, see
Lemma \ref{lemma-blow-up-normal-rational}.
\medskip\noindent
Ad D: The resolution of rational double points proceeds more or
less by hand, see
Section \ref{section-rational-double-points}.
A rational double point
is a hypersurface singularity (this is true but we don't prove it
as we don't need it). The local equation looks like
$$
a_{11} x_1^2 + a_{12} x_1x_2 + a_{13}x_1x_3 + a_{22} x_2^2 +
a_{23} x_2x_3 + a_{33} x_3^2 =
\sum a_{ijk} x_ix_jx_k
$$
Using that the quadratic part cannot be zero because the multiplicity
is $2$ and remains $2$ after any blowup and the fact that every blowup
is normal one quickly achieves a resolution. One twist is that we
do not have an invariant which decreases every blowup, but we rely
on the material on formal arcs from Section \ref{section-arcs}
to demonstrate that the process stops.
\medskip\noindent
To put everything together some additional work has
to be done. The main kink is that we want to lift a resolution
of the completion $A^\wedge$ to a resolution of $\Spec(A)$.
In order to do this we first show that if a resolution exists,
then there is a resolution by normalized blowups
(Lemma \ref{lemma-existence-implies-existence-by-normalized-blowing-ups}).
A sequence of normalized blowups can be lifted from the completion
by Lemma \ref{lemma-normalized-blowup-completion}.
We then use this even in the proof of resolution of complete
local rings $A$ because our strategy works by induction
on the degree of a finite inclusion $A_0 \subset A$ with
$A_0$ regular, see Lemma \ref{lemma-resolve-complete}.
With a stronger result in B (such as is proved in Lipman's paper)
this step could be avoided.
\section{A trace map in positive characteristic}
\label{section-trace}
\noindent
In this section $p$ will be a prime number. Let $R$ be an
$\mathbf{F}_p$-algebra. Given an $a \in R$ set $S = R[x]/(x^p - a)$.
Define an $R$-linear map
$$
\text{Tr}_x : \Omega_{S/R} \longrightarrow \Omega_R
$$
by the rule
$$
x^i\text{d}x \longmapsto
\left\{
\begin{matrix}
0 & \text{if} & 0 \leq i \leq p - 2, \\
\text{d}a & \text{if} & i = p - 1
\end{matrix}
\right.
$$
This makes sense as $\Omega_{S/R}$ is a free $R$-module with
basis $x^i\text{d}x$, $0 \leq i \leq p - 1$.
The following lemma implies that the trace map is well defined,
i.e., independent of the choice of the coordinate $x$.
\begin{lemma}
\label{lemma-trace-well-defined}
Let $\varphi : R[x]/(x^p - a) \to R[y]/(y^p - b)$ be an $R$-algebra
homomorphism. Then $\text{Tr}_x = \text{Tr}_y \circ \varphi$.
\end{lemma}
\begin{proof}
Say $\varphi(x) = \lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1}$
with $\lambda_i \in R$. The condition that mapping $x$ to
$\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1}$
induces an $R$-algebra homomorphism $R[x]/(x^p - a) \to R[y]/(y^p - b)$
is equivalent to the condition that
$$
a = \lambda_0^p + \lambda_1^p b + \ldots + \lambda_{p - 1}^pb^{p - 1}
$$
in the ring $R$. Consider the polynomial ring
$$
R_{univ} = \mathbf{F}_p[b, \lambda_0, \ldots, \lambda_{p - 1}]
$$
with the element
$a = \lambda_0^p + \lambda_1^p b + \ldots + \lambda_{p - 1}^pb^{p - 1}$
Consider the universal algebra map
$\varphi_{univ} : R_{univ}[x]/(x^p - a) \to R_{univ}[y]/(y^p - b)$
given by mapping $x$ to
$\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1}$.
We obtain a canonical map
$$
R_{univ} \longrightarrow R
$$
sending $b, \lambda_i$ to $b, \lambda_i$. By construction we get a
commutative diagram
$$
\xymatrix{
R_{univ}[x]/(x^p - a) \ar[r] \ar[d]_{\varphi_{univ}} &
R[x]/(x^p - a) \ar[d]^\varphi \\
R_{univ}[y]/(y^p - b) \ar[r] & R[y]/(y^p - b)
}
$$
and the horizontal arrows are compatible with the trace maps. Hence it
suffices to prove the lemma for the map $\varphi_{univ}$. Thus we may
assume $R = \mathbf{F}_p[b, \lambda_0, \ldots, \lambda_{p - 1}]$
is a polynomial ring. We will check the lemma holds in this case
by evaluating
$\text{Tr}_y(\varphi(x)^i\text{d}\varphi(x))$ for $i = 0 , \ldots, p - 1$.
\medskip\noindent
The case $0 \leq i \leq p - 2$. Expand
$$
(\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1})^i
(\lambda_1 + 2 \lambda_2 y + \ldots + (p - 1)\lambda_{p - 1}y^{p - 2})
$$
in the ring $R[y]/(y^p - b)$. We have to show that the coefficient
of $y^{p - 1}$ is zero. For this it suffices to show that
the expression above as a polynomial in $y$ has vanishing
coefficients in front of the powers $y^{pk - 1}$.
Then we write our polynomial as
$$
\frac{\text{d}}{(i + 1)\text{d}y}
(\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1})^{i + 1}
$$
and indeed the coefficients of $y^{kp - 1}$ are all zero.
\medskip\noindent
The case $i = p - 1$. Expand
$$
(\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1})^{p - 1}
(\lambda_1 + 2 \lambda_2 y + \ldots + (p - 1)\lambda_{p - 1}y^{p - 2})
$$
in the ring $R[y]/(y^p - b)$. To finish the proof we have to show that
the coefficient of $y^{p - 1}$ times $\text{d}b$ is $\text{d}a$.
Here we use that $R$ is $S/pS$ where
$S = \mathbf{Z}[b, \lambda_0, \ldots, \lambda_{p - 1}]$.
Then the above, as a polynomial in $y$, is equal to
$$
\frac{\text{d}}{p\text{d}y}
(\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1})^p
$$
Since $\frac{\text{d}}{\text{d}y}(y^{pk}) = pk y^{pk - 1}$
it suffices to understand the coefficients of $y^{pk}$ in the polynomial
$(\lambda_0 + \lambda_1 y + \ldots + \lambda_{p - 1}y^{p - 1})^p$
modulo $p$. The sum of these terms gives
$$
\lambda_0^p + \lambda_1^py^p + \ldots + \lambda_{p - 1}^py^{p(p - 1)}
\bmod p
$$
Whence we see that we obtain after applying the operator
$\frac{\text{d}}{p\text{d}y}$ and after reducing modulo $y^p - b$
the value
$$
\lambda_1^p + 2\lambda_2^pb + \ldots + (p - 1)\lambda_{p - 1}b^{p - 2}
$$
for the coefficient of $y^{p - 1}$ we wanted to compute. Now because
$a = \lambda_0^p + \lambda_1^p b + \ldots + \lambda_{p - 1}^pb^{p - 1}$
in $R$ we obtain that
$$
\text{d}a = (\lambda_1^p + 2 \lambda_2^p b + \ldots +
(p - 1) \lambda_{p - 1}^p b^{p - 2}) \text{d}b
$$
in $R$. This proves that the coefficient of $y^{p - 1}$ is as desired.
\end{proof}
\begin{lemma}
\label{lemma-trace-higher}
Let $\mathbf{F}_p \subset \Lambda \subset R \subset S$ be ring extensions
and assume that $S$ is isomorphic to $R[x]/(x^p - a)$ for some $a \in R$.
Then there are canonical $R$-linear maps
$$
\text{Tr} :
\Omega^{t + 1}_{S/\Lambda}
\longrightarrow
\Omega_{R/\Lambda}^{t + 1}
$$
for $t \geq 0$ such that
$$
\eta_1 \wedge \ldots \wedge \eta_t \wedge x^i\text{d}x
\longmapsto
\left\{
\begin{matrix}
0 & \text{if} & 0 \leq i \leq p - 2, \\
\eta_1 \wedge \ldots \wedge \eta_t \wedge \text{d}a & \text{if} & i = p - 1
\end{matrix}
\right.
$$
for $\eta_i \in \Omega_{R/\Lambda}$ and such that $\text{Tr}$ annihilates the
image of
$S \otimes_R \Omega_{R/\Lambda}^{t + 1} \to \Omega_{S/\Lambda}^{t + 1}$.
\end{lemma}
\begin{proof}
For $t = 0$ we use the composition
$$
\Omega_{S/\Lambda} \to \Omega_{S/R} \to \Omega_R \to \Omega_{R/\Lambda}
$$
where the second map is Lemma \ref{lemma-trace-well-defined}.
There is an exact sequence
$$
H_1(L_{S/R}) \xrightarrow{\delta} \Omega_{R/\Lambda} \otimes_R S \to
\Omega_{S/\Lambda} \to \Omega_{S/R} \to 0
$$
(Algebra, Lemma \ref{algebra-lemma-exact-sequence-NL}).
The module $\Omega_{S/R}$ is free over $S$ with basis $\text{d}x$
and the module $H^1(L_{S/R})$ is free over $S$ with basis $x^p - a$
which $\delta$ maps to $-\text{d}a \otimes 1$ in
$\Omega_{R/\Lambda} \otimes_R S$. In particular, if we set
$$
M = \Coker(R \to \Omega_{R/\Lambda}, 1 \mapsto -\text{d}a)
$$
then we see that $\Coker(\delta) = M \otimes_R S$. We obtain a
canonical map
$$
\Omega^{t + 1}_{S/\Lambda} \to
\wedge_S^t(\Coker(\delta)) \otimes_S \Omega_{S/R} =
\wedge^t_R(M) \otimes_R \Omega_{S/R}
$$
Now, since the image of the map
$\text{Tr} : \Omega_{S/R} \to \Omega_{R/\lambda}$
of Lemma \ref{lemma-trace-well-defined} is contained in $R\text{d}a$ we
see that wedging with an element in the image annihilates $\text{d}a$.
Hence there is a canonical map
$$
\wedge^t_R(M) \otimes_R \Omega_{S/R} \to \Omega_{R/\Lambda}^{t + 1}
$$
mapping
$\overline{\eta}_1 \wedge \ldots \wedge \overline{\eta}_t \wedge \omega$
to $\eta_1 \wedge \ldots \wedge \eta_t \wedge \text{Tr}(\omega)$.
\end{proof}
\begin{lemma}
\label{lemma-trace-extends}
Let $S$ be a scheme over $\mathbf{F}_p$. Let $f : Y \to X$ be a finite morphism
of Noetherian normal integral schemes over $S$. Assume
\begin{enumerate}
\item the extension of function fields is purely inseparable of degree $p$, and
\item $\Omega_{X/S}$ is a coherent $\mathcal{O}_X$-module (for example
if $X$ is of finite type over $S$).
\end{enumerate}
For $i \geq 1$ there is a canonical map
$$
\text{Tr} : f_*\Omega^i_{Y/S} \longrightarrow (\Omega_{X/S}^i)^{**}
$$
whose stalk in the generic point of $X$ recovers the trace map of
Lemma \ref{lemma-trace-higher}.
\end{lemma}
\begin{proof}
The exact sequence $f^*\Omega_{X/S} \to \Omega_{Y/S} \to \Omega_{Y/X} \to 0$
shows that $\Omega_{Y/S}$ and hence $f_*\Omega_{Y/S}$ are coherent modules
as well. Thus it suffices to prove the trace map in the generic point
extends to stalks at $x \in X$ with $\dim(\mathcal{O}_{X, x}) = 1$, see
Divisors, Lemma \ref{divisors-lemma-describe-reflexive-hull}.
Thus we reduce to the case discussed in the next paragraph.
\medskip\noindent
Assume $X = \Spec(A)$ and $Y = \Spec(B)$ with $A$ a discrete valuation
ring and $B$ finite over $A$. Since the induced extension $K \subset L$
of fraction fields is purely inseparable, we see that $B$ is local too.
Hence $B$ is a discrete valuation ring too. Then either
\begin{enumerate}
\item $B/A$ has ramification index $p$ and hence $B = A[x]/(x^p - a)$
where $a \in A$ is a uniformizer, or
\item $\mathfrak m_B = \mathfrak m_A B$ and the residue field
$B/\mathfrak m_A B$ is purely inseparable of degree $p$ over
$\kappa_A = A/\mathfrak m_A$.
Choose any $x \in B$ whose residue class is not in $\kappa_A$
and then we'll have $B = A[x]/(x^p - a)$ where $a \in A$ is
a unit.
\end{enumerate}
Let $\Spec(\Lambda) \subset S$ be an affine open such that
$X$ maps into $\Spec(\Lambda)$. Then we can apply
Lemma \ref{lemma-trace-higher}
to see that the trace map extends to
$\Omega^i_{B/\Lambda} \to \Omega^i_{A/\Lambda}$
for all $i \geq 1$.
\end{proof}
\section{Quadratic transformations}
\label{section-quadratic}
\noindent
In this section we study what happens when we blow up a nonsingular point
on a surface. We hesitate the formally define such a morphism as a
{\it quadratic transformation} as on the one hand often other names are
used and on the other hand the phrase ``quadratic transformation'' is
sometimes used with a different meaning.
\begin{lemma}
\label{lemma-blowup}
Let $(A, \mathfrak m, \kappa)$ be a regular local ring of dimension $2$.
Let $f : X \to S = \Spec(A)$ be the blowing up of $A$ in $\mathfrak m$.
There is a closed immersion
$$
r : X \longrightarrow \mathbf{P}^1_S
$$
over $S$ such that
\begin{enumerate}
\item $r|_E : E \to \mathbf{P}^1_\kappa$ is an isomorphism,
\item $\mathcal{O}_X(E) = \mathcal{O}_X(-1) =
r^*\mathcal{O}_{\mathbf{P}^1}(-1)$, and
\item $\mathcal{C}_{E/X} = (r|_E)^*\mathcal{O}_{\mathbf{P}^1}(1)$ and
$\mathcal{N}_{E/X} = (r|_E)^*\mathcal{O}_{\mathbf{P}^1}(-1)$.
\end{enumerate}
\end{lemma}
\begin{proof}
As $A$ is regular of dimension $2$ we can write $\mathfrak m = (x, y)$.
Then $x$ and $y$ placed in degree $1$ generate the Rees algebra
$\bigoplus_{n \geq 0} \mathfrak m^n$ over $A$. Recall that
$X = \text{Proj}(\bigoplus_{n \geq 0} \mathfrak m^n)$, see
Divisors, Lemma \ref{divisors-lemma-blowing-up-affine}.
Thus the surjection
$$
A[T_0, T_1] \longrightarrow \bigoplus\nolimits_{n \geq 0} \mathfrak m^n,
\quad
T_0 \mapsto x,\ T_1 \mapsto y
$$
of graded $A$-algebras induces a closed immersion
$r : X \to \mathbf{P}^1_S = \text{Proj}(A[T_0, T_1])$
such that $\mathcal{O}_X(1) = r^*\mathcal{O}_{\mathbf{P}^1_S}(1)$, see
Constructions, Lemma
\ref{constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj}.
This proves (2) because $\mathcal{O}_X(E) = \mathcal{O}_X(-1)$
by Divisors, Lemma
\ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}.
\medskip\noindent
To prove (1) note that
$$
\left(\bigoplus\nolimits_{n \geq 0} \mathfrak m^n\right) \otimes_A \kappa =
\bigoplus\nolimits_{n \geq 0} \mathfrak m^n/\mathfrak m^{n + 1} \cong
\kappa[\overline{x}, \overline{y}]
$$
a polynomial algebra, see Algebra, Lemma \ref{algebra-lemma-regular-graded}.
This proves that the fibre of $X \to S$ over $\Spec(\kappa)$ is equal to
$\text{Proj}(\kappa[\overline{x}, \overline{y}]) = \mathbf{P}^1_\kappa$, see
Constructions, Lemma \ref{constructions-lemma-base-change-map-proj}.
Recall that $E$ is the closed subscheme of $X$ defined by
$\mathfrak m\mathcal{O}_X$, i.e., $E = X_\kappa$.
By our choice of the morphism $r$ we see that $r|_E$ in fact
produces the identification of $E = X_\kappa$ with the special
fibre of $\mathbf{P}^1_S \to S$.
\medskip\noindent
Part (3) follows from (1) and (2) and Divisors, Lemma
\ref{divisors-lemma-conormal-effective-Cartier-divisor}.
\end{proof}
\begin{lemma}
\label{lemma-blowup-regular}
Let $(A, \mathfrak m, \kappa)$ be a regular local ring of dimension $2$.
Let $f : X \to S = \Spec(A)$ be the blowing up of $A$ in $\mathfrak m$.
Then $X$ is an irreducible regular scheme.
\end{lemma}
\begin{proof}
Observe that $X$ is integral by
Divisors, Lemma \ref{divisors-lemma-blow-up-integral-scheme}
and
Algebra, Lemma \ref{algebra-lemma-regular-domain}.
To see $X$ is regular it suffices to check that $\mathcal{O}_{X, x}$
is regular for closed points $x \in X$, see
Properties, Lemma \ref{properties-lemma-characterize-regular}.
Let $x \in X$ be a closed point. Since $f$ is proper $x$ maps to
$\mathfrak m$, i.e., $x$ is a point of the exceptional divisor $E$.
Then $E$ is an effective Cartier divisor and $E \cong \mathbf{P}^1_\kappa$.
Thus if $f \in \mathfrak m_x \subset \mathcal{O}_{X, x}$ is a local
equation for $E$, then
$\mathcal{O}_{X, x}/(f) \cong \mathcal{O}_{\mathbf{P}^1_\kappa, x}$.
Since $\mathbf{P}^1_\kappa$ is covered by two affine opens which are the
spectrum of a polynomial ring over $\kappa$, we see that
$\mathcal{O}_{\mathbf{P}^1_\kappa, x}$ is regular by
Algebra, Lemma \ref{algebra-lemma-dim-affine-space}.
We conclude by
Algebra, Lemma \ref{algebra-lemma-regular-mod-x}.
\end{proof}
\begin{lemma}
\label{lemma-blowup-pic}
Let $(A, \mathfrak m, \kappa)$ be a regular local ring of dimension $2$.
Let $f : X \to S = \Spec(A)$ be the blowing up of $A$ in $\mathfrak m$.
Then $\text{Pic}(X) = \mathbf{Z}$ generated by $\mathcal{O}_X(E)$.
\end{lemma}
\begin{proof}
Recall that $E = \mathbf{P}^1_\kappa$ has Picard group $\mathbf{Z}$
with generator $\mathcal{O}(1)$, see
Divisors, Lemma \ref{divisors-lemma-Pic-projective-space-UFD}.
By Lemma \ref{lemma-blowup} the invertible $\mathcal{O}_X$-module
$\mathcal{O}_X(E)$ restricts to $\mathcal{O}(-1)$. Hence
$\mathcal{O}_X(E)$ generates an infinite cyclic group in $\text{Pic}(X)$.
Since $A$ is regular it is a UFD, see More on Algebra,
Lemma \ref{more-algebra-lemma-regular-local-UFD}.
Then the punctured spectrum $U = S \setminus \{\mathfrak m\} = X \setminus E$
has trivial Picard group, see
Divisors, Lemma \ref{divisors-lemma-open-subscheme-UFD}.
Hence for every invertible $\mathcal{O}_X$-module $\mathcal{L}$
there is an isomorphism $s : \mathcal{O}_U \to \mathcal{L}|_U$.
Then $s$ is a regular meromorphic section of $\mathcal{L}$
and we see that $\text{div}_\mathcal{L}(s) = nE$ for some
$n \in \mathbf{Z}$
(Divisors, Definition \ref{divisors-definition-divisor-invertible-sheaf}).
By Divisors, Lemma \ref{divisors-lemma-normal-c1-injective}
(and the fact that $X$ is normal by Lemma \ref{lemma-blowup-regular})
we conclude that $\mathcal{L} = \mathcal{O}_X(nE)$.
\end{proof}
\begin{lemma}
\label{lemma-cohomology-of-blowup}
Let $(A, \mathfrak m, \kappa)$ be a regular local ring of dimension $2$.
Let $f : X \to S = \Spec(A)$ be the blowing up of $A$ in $\mathfrak m$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
\begin{enumerate}
\item $H^p(X, \mathcal{F}) = 0$ for $p \not \in \{0, 1\}$,
\item $H^1(X, \mathcal{O}_X(n)) = 0$ for $n \geq -1$,
\item $H^1(X, \mathcal{F}) = 0$ if $\mathcal{F}$ or $\mathcal{F}(1)$
is globally generated,
\item $H^0(X, \mathcal{O}_X(n)) = \mathfrak m^{\max(0, n)}$,
\item $\text{length}_A H^1(X, \mathcal{O}_X(n)) = -n(-n - 1)/2$
if $n < 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
If $\mathfrak m = (x, y)$, then $X$ is covered by the spectra
of the affine blowup algebras $A[\frac{\mathfrak m}{x}]$ and
$A[\frac{\mathfrak m}{y}]$ because $x$ and $y$ placed in degree $1$
generate the Rees algebra $\bigoplus \mathfrak m^n$ over $A$.
See Divisors, Lemma \ref{divisors-lemma-blowing-up-affine} and
Constructions, Lemma \ref{constructions-lemma-proj-quasi-compact}.
Since $X$ is separated by
Constructions, Lemma \ref{constructions-lemma-proj-separated}
we see that cohomology of quasi-coherent sheaves vanishes in
degrees $\geq 2$ by Cohomology of Schemes, Lemma
\ref{coherent-lemma-vanishing-nr-affines}.
\medskip\noindent
Let $i : E \to X$ be the exceptional divisor, see
Divisors, Definition \ref{divisors-definition-blow-up}.
Recall that $\mathcal{O}_X(-E) = \mathcal{O}_X(1)$ is
$f$-relatively ample, see
Divisors, Lemma \ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}.
Hence we know that $H^1(X, \mathcal{O}_X(-nE)) = 0$ for some $n > 0$,
see Cohomology of Schemes, Lemma \ref{coherent-lemma-kill-by-twisting}.
Consider the filtration
$$
\mathcal{O}_X(-nE) \subset \mathcal{O}_X(-(n - 1)E) \subset
\ldots \subset \mathcal{O}_X(-E) \subset \mathcal{O}_X \subset \mathcal{O}_X(E)
$$
The successive quotients are the sheaves
$$
\mathcal{O}_X(-t E)/\mathcal{O}_X(-(t + 1)E) =
\mathcal{O}_X(t)/\mathcal{I}(t) =
i_*\mathcal{O}_E(t)
$$
where $\mathcal{I} = \mathcal{O}_X(-E)$ is the ideal sheaf of $E$.
By Lemma \ref{lemma-blowup} we have $E = \mathbf{P}^1_\kappa$ and
$\mathcal{O}_E(1)$ indeed corresponds to the usual Serre twist of
the structure sheaf on $\mathbf{P}^1$. Hence the cohomology
of $\mathcal{O}_E(t)$ vanishes in degree $1$ for $t \geq -1$, see
Cohomology of Schemes, Lemma
\ref{coherent-lemma-cohomology-projective-space-over-ring}.
Since this is equal to $H^1(X, i_*\mathcal{O}_E(t))$ (by
Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-cohomology})
we find that $H^1(X, \mathcal{O}_X(-(t + 1)E)) \to H^1(X, \mathcal{O}_X(-tE))$
is surjective for $t \geq -1$. Hence
$$
0 = H^1(X, \mathcal{O}_X(-nE))
\longrightarrow
H^1(X, \mathcal{O}_X(-tE)) = H^1(X, \mathcal{O}_X(t))
$$
is surjective for $t \geq -1$ which proves (2).
\medskip\noindent
Let $\mathcal{F}$ be globally generated. This means there exists
a short exact sequence
$$
0 \to \mathcal{G} \to \bigoplus\nolimits_{i \in I} \mathcal{O}_X
\to \mathcal{F} \to 0
$$
Note that $H^1(X, \bigoplus_{i \in I} \mathcal{O}_X) =
\bigoplus_{i \in I} H^1(X, \mathcal{O}_X)$ by
Cohomology, Lemma \ref{cohomology-lemma-quasi-separated-cohomology-colimit}.
By part (2) we have $H^1(X, \mathcal{O}_X) = 0$.
If $\mathcal{F}(1)$ is globally generated, then we can find a
surjection $\bigoplus_{i \in I} \mathcal{O}_X(-1) \to \mathcal{F}$
and argue in a similar fashion.
In other words, part (3) follows from part (2).
\medskip\noindent
For part (4) we note that for all $n$ large enough we have
$\Gamma(X, \mathcal{O}_X(n)) = \mathfrak m^n$, see
Cohomology of Schemes, Lemma \ref{coherent-lemma-recover-tail-graded-module}.
If $n \geq 0$, then we can use the short exact sequence
$$
0 \to \mathcal{O}_X(n) \to \mathcal{O}_X(n - 1) \to
i_*\mathcal{O}_E(n - 1) \to 0
$$
and the vanishing of $H^1$ for the sheaf on the left to get a commutative
diagram
$$
\xymatrix{
0 \ar[r] &
\mathfrak m^{\max(0, n)} \ar[r] \ar[d] &
\mathfrak m^{\max(0, n - 1)} \ar[r] \ar[d] &
\mathfrak m^{\max(0, n)}/\mathfrak m^{\max(0, n - 1)} \ar[r] \ar[d] & 0\\
0 \ar[r] &
\Gamma(X, \mathcal{O}_X(n)) \ar[r] &
\Gamma(X, \mathcal{O}_X(n - 1)) \ar[r] &
\Gamma(E, \mathcal{O}_E(n - 1)) \ar[r] & 0
}
$$
with exact rows. In fact, the rows are exact also for $n < 0$
because in this case the groups on the right are zero.
In the proof of Lemma \ref{lemma-blowup}
we have seen that the right vertical arrow is an isomorphism
(details omitted). Hence if the left vertical arrow is an isomorphism, so
is the middle one. In this way we see that (4) holds by
descending induction on $n$.
\medskip\noindent
Finally, we prove (5) by descending induction on $n$ and the sequences
$$
0 \to \mathcal{O}_X(n) \to \mathcal{O}_X(n - 1) \to
i_*\mathcal{O}_E(n - 1) \to 0
$$
Namely, for $n \geq -1$ we already know $H^1(X, \mathcal{O}_X(n)) = 0$.
Since
$$
H^1(X, i_*\mathcal{O}_E(-2)) =
H^1(E, \mathcal{O}_E(-2)) =
H^1(\mathbf{P}^1_\kappa, \mathcal{O}(-2)) \cong \kappa
$$
by Cohomology of Schemes, Lemma
\ref{coherent-lemma-cohomology-projective-space-over-ring}
which has length $1$ as an $A$-module, we conclude from the long exact
cohomology sequence that (5) holds for $n = -2$. And so on and so forth.
\end{proof}
\begin{lemma}
\label{lemma-blowup-improve}
Let $(A, \mathfrak m)$ be a regular local ring of dimension $2$.
Let $f : X \to S = \Spec(A)$ be the blowing up of $A$ in $\mathfrak m$.
Let $\mathfrak m^n \subset I \subset \mathfrak m$ be an ideal.
Let $d \geq 0$ be the largest integer such that
$$
I \mathcal{O}_X \subset \mathcal{O}_X(-dE)
$$
where $E$ is the exceptional divisor. Set
$\mathcal{I}' = I\mathcal{O}_X(dE) \subset \mathcal{O}_X$.
Then $d > 0$, the sheaf
$\mathcal{O}_X/\mathcal{I}'$ is supported in finitely many
closed points $x_1, \ldots, x_r$ of $X$, and
\begin{align*}
\text{length}_A(A/I)
& >
\text{length}_A \Gamma(X, \mathcal{O}_X/\mathcal{I}') \\
& \geq
\sum\nolimits_{i = 1, \ldots, r}
\text{length}_{\mathcal{O}_{X, x_i}}
(\mathcal{O}_{X, x_i}/\mathcal{I}'_{x_i})
\end{align*}
\end{lemma}
\begin{proof}
Since $I \subset \mathfrak m$ we see that every element of $I$
vanishes on $E$. Thus we see that $d \geq 1$. On the other hand, since
$\mathfrak m^n \subset I$ we see that $d \leq n$. Consider the
short exact sequence
$$
0 \to I\mathcal{O}_X \to \mathcal{O}_X \to \mathcal{O}_X/I\mathcal{O}_X \to 0
$$
Since $I\mathcal{O}_X$ is globally generated, we see that
$H^1(X, I\mathcal{O}_X) = 0$ by Lemma \ref{lemma-cohomology-of-blowup}.
Hence we obtain a surjection
$A/I \to \Gamma(X, \mathcal{O}_X/I\mathcal{O}_X)$. Consider the short exact
sequence
$$
0 \to
\mathcal{O}_X(-dE)/I\mathcal{O}_X \to
\mathcal{O}_X/I\mathcal{O}_X \to
\mathcal{O}_X/\mathcal{O}_X(-dE) \to 0
$$
By Divisors, Lemma \ref{divisors-lemma-codim-1-part}
we see that $\mathcal{O}_X(-dE)/I\mathcal{O}_X$ is supported in finitely many
closed points of $X$. In particular, this coherent sheaf has vanishing higher
cohomology groups (detail omitted). Thus in the following diagram
$$
\xymatrix{
& & A/I \ar[d] \\
0 \ar[r] &
\Gamma(X, \mathcal{O}_X(-dE)/I\mathcal{O}_X) \ar[r] &
\Gamma(X, \mathcal{O}_X/I\mathcal{O}_X) \ar[r] &
\Gamma(X, \mathcal{O}_X/\mathcal{O}_X(-dE)) \ar[r] & 0
}
$$
the bottom row is exact and the vertical arrow surjective. We have
$$
\text{length}_A \Gamma(X, \mathcal{O}_X(-dE)/I\mathcal{O}_X) <
\text{length}_A(A/I)
$$
since $\Gamma(X, \mathcal{O}_X/\mathcal{O}_X(-dE))$ is nonzero.
Namely, the image of $1 \in \Gamma(X, \mathcal{O}_X)$
is nonzero as $d > 0$.
\medskip\noindent
To finish the proof we translate the results above into the statements
of the lemma. Since
$\mathcal{O}_X(dE)$ is invertible we have
$$
\mathcal{O}_X/\mathcal{I}' =
\mathcal{O}_X(-dE)/I\mathcal{O}_X \otimes_{\mathcal{O}_X} \mathcal{O}_X(dE).
$$
Thus $\mathcal{O}_X/\mathcal{I}'$ and $\mathcal{O}_X(-dE)/I\mathcal{O}_X$
are supported in the same set of finitely many
closed points, say $x_1, \ldots, x_r \in E \subset X$.
Moreover we obtain
$$
\Gamma(X, \mathcal{O}_X(-dE)/I\mathcal{O}_X) =
\bigoplus \mathcal{O}_X(-dE)_{x_i}/I\mathcal{O}_{X, x_i}
\cong
\bigoplus \mathcal{O}_{X, x_i}/\mathcal{I}'_{x_i} =
\Gamma(X, \mathcal{O}_X/\mathcal{I}')
$$
because an invertible module over a local ring is trivial.
Thus we obtain the strict inequality. We also get the second because
$$
\text{length}_A(\mathcal{O}_{X, x_i}/\mathcal{I}'_{x_i}) \geq
\text{length}_{\mathcal{O}_{X, x_i}}(\mathcal{O}_{X, x_i}/\mathcal{I}'_{x_i})
$$
as is immediate from the definition of length.
\end{proof}
\begin{lemma}
\label{lemma-differentials-of-blowup}
Let $(A, \mathfrak m, \kappa)$ be a regular local ring of dimension $2$.
Let $f : X \to S = \Spec(A)$ be the blowing up of $A$ in $\mathfrak m$.
Then $\Omega_{X/S} = i_*\Omega_{E/\kappa}$, where $i : E \to X$
is the immersion of the exceptional divisor.
\end{lemma}
\begin{proof}
Writing $\mathbf{P}^1 = \mathbf{P}^1_S$, let
$r : X \to \mathbf{P}^1$ be as in Lemma \ref{lemma-blowup}.
Then we have an exact sequence
$$
\mathcal{C}_{X/\mathbf{P}^1} \to r^*\Omega_{\mathbf{P}^1/S} \to
\Omega_{X/S} \to 0
$$
see Morphisms, Lemma \ref{morphisms-lemma-differentials-relative-immersion}.
Since $\Omega_{\mathbf{P}^1/S}|_E = \Omega_{E/\kappa}$ by
Morphisms, Lemma \ref{morphisms-lemma-base-change-differentials}
it suffices to see that the first arrow defines a surjection
onto the kernel of the canonical map
$r^*\Omega_{\mathbf{P}^1/S} \to i_*\Omega_{E/\kappa}$.
This we can do locally. With notation as in the proof of
Lemma \ref{lemma-blowup} on an affine open of $X$ the morphism $f$
corresponds to the ring map
$$
A \to A[t]/(xt - y)
$$
where $x, y \in \mathfrak m$ are generators. Thus
$\text{d}(xt - y) = x\text{d}t$ and $y\text{d}t = t \cdot x \text{d}t$
which proves what we want.
\end{proof}
\section{Dominating by quadratic transformations}
\label{section-dominating-by-quadratic}
\noindent
Using the result above we can prove that blowups in points dominate
any modification of a regular $2$ dimensional scheme.
\medskip\noindent
Let $X$ be a scheme. Let $x \in X$ be a closed point. As usual, we view
$i : x = \Spec(\kappa(x)) \to X$ as a closed subscheme.
The {\it blowing up $X' \to X$ of $X$ at $x$} is the blowing up of $X$
in the closed subscheme $x \subset X$. Observe that if $X$ is locally
Noetherian, then $X' \to X$ is projective (in particular proper) by
Divisors, Lemma \ref{divisors-lemma-blowing-up-projective}.
\begin{lemma}
\label{lemma-make-ideal-principal}
Let $X$ be a Noetherian scheme. Let $T \subset X$ be a finite set of
closed points $x$ such that $\mathcal{O}_{X, x}$ is
regular of dimension $2$ for $x \in T$.
Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent
sheaf of ideals such that $\mathcal{O}_X/\mathcal{I}$ is supported
on $T$.
Then there exists a sequence
$$
X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X
$$
where $X_{i + 1} \to X_i$ is the blowing up of $X_i$ at a closed
point $x_i$ lying above a point of $T$ such that
$\mathcal{I}\mathcal{O}_{X_n}$ is an invertible ideal sheaf.
\end{lemma}
\begin{proof}
Say $T = \{x_1, \ldots, x_r\}$. Set
$$
n_i = \text{length}_{\mathcal{O}_{X, x_i}}(\mathcal{O}_{X, x_i}/I_i)
$$
This is finite as $\mathcal{O}_X/\mathcal{I}$ is supported on $T$
and hence $\mathcal{O}_{X, x_i}/I_i$ has support equal to
$\{\mathfrak m_{x_i}\}$ (see Algebra, Lemma \ref{algebra-lemma-support-point}).
We are going to use induction on $\sum n_i$. If $n_i = 0$ for all
$i$, then $\mathcal{I} = \mathcal{O}_X$ and we are done.
\medskip\noindent
Suppose $n_i > 0$. Let $X' \to X$ be the blowing up of $X$ in $x_i$
(see discussion above the lemma).
Since $\Spec(\mathcal{O}_{X, x_i}) \to X$ is flat we see that
$X' \times_X \Spec(\mathcal{O}_{X, x_i})$ is the blowup of
the ring $\mathcal{O}_{X, x_i}$ in the maximal ideal, see
Divisors, Lemma
\ref{divisors-lemma-flat-base-change-blowing-up}.
Hence the square in the commutative diagram
$$
\xymatrix{
\text{Proj}(\bigoplus\nolimits_{d \geq 0} \mathfrak m_{x_i}^d) \ar[r] \ar[d] &
X' \ar[d] \\
\Spec(\mathcal{O}_{X, x_i}) \ar[r] & X
}
$$
is cartesian. Let $E \subset X'$ and
$E' \subset \text{Proj}(\bigoplus\nolimits_{d \geq 0} \mathfrak m_{x_i}^d)$
be the exceptional divisors. Let $d \geq 1$ be the integer found in
Lemma \ref{lemma-blowup-improve} for the ideal
$\mathcal{I}_i \subset \mathcal{O}_{X, x_i}$.
Since the horizontal arrows in the diagram are flat, since
$E' \to E$ is surjective, and since $E'$ is the pullback of $E$, we see that
$$
\mathcal{I}\mathcal{O}_{X'} \subset \mathcal{O}_{X'}(-dE)
$$
(some details omitted).
Set $\mathcal{I}' = \mathcal{I}\mathcal{O}_{X'}(dE) \subset \mathcal{O}_{X'}$.
Then we see that $\mathcal{O}_{X'}/\mathcal{I}'$ is supported in finitely
many closed points $T' \subset |X'|$ because this holds over
$X \setminus \{x_i\}$ and for the pullback to
$\text{Proj}(\bigoplus\nolimits_{d \geq 0} \mathfrak m_{x_i}^d)$.
The final assertion of Lemma \ref{lemma-blowup-improve}
tells us that the sum of the lengths of the stalks
$\mathcal{O}_{X', x'}/\mathcal{I}'\mathcal{O}_{X', x'}$
for $x'$ lying over $x_i$ is $< n_i$. Hence the sum of the lengths
has decreased.
\medskip\noindent
By induction hypothesis, there exists a sequence
$$
X'_n \to \ldots \to X'_1 \to X'
$$
of blowups at closed points lying over $T'$ such that
$\mathcal{I}'\mathcal{O}_{X'_n}$ is invertible. Since
$\mathcal{I}'\mathcal{O}_{X'}(-dE) = \mathcal{I}\mathcal{O}_{X'}$, we see
that $\mathcal{I}\mathcal{O}_{X'_n} =
\mathcal{I}'\mathcal{O}_{X'_n}(-d(f')^{-1}E)$
where $f' : X'_n \to X'$ is the composition.
Note that $(f')^{-1}E$ is an effective Cartier divisor by
Divisors, Lemma \ref{divisors-lemma-blow-up-pullback-effective-Cartier}.
Thus we are done by
Divisors, Lemma \ref{divisors-lemma-sum-effective-Cartier-divisors}.
\end{proof}
\begin{lemma}
\label{lemma-dominate-by-blowing-up-in-points}
Let $X$ be a Noetherian scheme. Let $T \subset X$ be a finite set of
closed points $x$ such that $\mathcal{O}_{X, x}$ is a regular local
ring of dimension $2$. Let $f : Y \to X$ be a proper morphism of
schemes which is an isomorphism over $U = X \setminus T$.
Then there exists a sequence
$$
X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X
$$
where $X_{i + 1} \to X_i$ is the blowing up of $X_i$ at a closed
point $x_i$ lying above a point of $T$ and a factorization $X_n \to Y \to X$
of the composition.
\end{lemma}
\begin{proof}
By More on Flatness, Lemma \ref{flat-lemma-dominate-modification-by-blowup}
there exists a $U$-admissible blowup $X' \to X$ which dominates
$Y \to X$. Hence we may assume there exists an ideal sheaf
$\mathcal{I} \subset \mathcal{O}_X$ such that
$\mathcal{O}_X/\mathcal{I}$ is supported on $T$ and such that
$Y$ is the blowing up of $X$ in $\mathcal{I}$.
By Lemma \ref{lemma-make-ideal-principal}
there exists a sequence
$$
X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X
$$
where $X_{i + 1} \to X_i$ is the blowing up of $X_i$ at a closed
point $x_i$ lying above a point of $T$ such that
$\mathcal{I}\mathcal{O}_{X_n}$ is an invertible ideal sheaf.
By the universal property of blowing up
(Divisors, Lemma
\ref{divisors-lemma-universal-property-blowing-up})
we find the desired factorization.
\end{proof}
\begin{lemma}
\label{lemma-extend-rational-map-blowing-up}
Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is
regular and has dimension $2$. Let $Y$ be a proper
scheme over $S$. Given an $S$-rational map $f : U \to Y$ from
$X$ to $Y$ there exists a sequence
$$
X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X
$$
and an $S$-morphism $f_n : X_n \to Y$ such that $X_{i + 1} \to X_i$
blowing up of $X_i$ at a closed point not lying over $U$
and $f_n$ and $f$ agree.
\end{lemma}
\begin{proof}
We may assume $U$ contains every point of codimension $1$, see
Morphisms, Lemma \ref{morphisms-lemma-extend-across}.
Hence the complement $T \subset X$ of $U$ is a finite set
of closed points whose local rings are regular of dimension $2$.
Applying
Divisors, Lemma \ref{divisors-lemma-extend-rational-map-after-modification}
we find a proper morphism $p : X' \to X$ which is an isomorphism
over $U$ and a morphism $f' : X' \to Y$ agreeing with $f$ over $U$.
Apply Lemma \ref{lemma-dominate-by-blowing-up-in-points}
to the morphism $p : X' \to X$. The composition $X_n \to X' \to Y$ is
the desired morphism.
\end{proof}
\section{Dominating by normalized blowups}
\label{section-normalized-blowups}
\noindent
In this section we prove that a modification of a surface can be dominated
by a sequence of normalized blowups in points.
\begin{definition}
\label{definition-normalized-blowup}
Let $X$ be a scheme such that every quasi-compact open has finitely
many irreducible components. Let $x \in X$ be a closed point.
The {\it normalized blowup of $X$ at $x$} is the composition
$X'' \to X' \to X$ where $X' \to X$ is the blowup
of $X$ in $x$ and $X'' \to X'$ is the normalization of $X'$.
\end{definition}
\noindent
Here the normalization $X'' \to X'$ is defined as the scheme $X'$
has an open covering by opens which have finitely many irreducible
components by
Divisors, Lemma \ref{divisors-lemma-blow-up-and-irreducible-components}.
See Morphisms, Definition \ref{morphisms-definition-normalization}
for the definition of the normalization.
\medskip\noindent
In general the normalized blowing up need not be proper even
when $X$ is Noetherian. Recall that a scheme is Nagata if it
has an open covering by affines which are spectra of Nagata rings
(Properties, Definition \ref{properties-definition-nagata}).
\begin{lemma}
\label{lemma-Nagata-normalized-blowup}
In Definition \ref{definition-normalized-blowup} if $X$ is Nagata,
then the normalized blowing up of $X$ at $x$ is
normal, Nagata, and proper over $X$.
\end{lemma}
\begin{proof}
The blowup morphism $X' \to X$ is proper
(as $X$ is locally Noetherian we may apply
Divisors, Lemma \ref{divisors-lemma-blowing-up-projective}).
Thus $X'$ is Nagata
(Morphisms, Lemma \ref{morphisms-lemma-finite-type-nagata}).
Therefore the normalization $X'' \to X'$ is finite
(Morphisms, Lemma \ref{morphisms-lemma-nagata-normalization})
and we conclude that $X'' \to X$ is proper as well
(Morphisms, Lemmas \ref{morphisms-lemma-finite-proper} and
\ref{morphisms-lemma-composition-proper}).
It follows that the normalized blowing up
is a normal (Morphisms, Lemma
\ref{morphisms-lemma-normalization-normal})
Nagata algebraic space.
\end{proof}
\noindent
In the following lemma we need to assume $X$ is Noetherian in order
to make sure that it has finitely many irreducible components.
Then the properness of $f : Y \to X$ assures that $Y$ has finitely
many irreducible components too and it makes sense to require
$f$ to be birational
(Morphisms, Definition \ref{morphisms-definition-birational}).
\begin{lemma}
\label{lemma-dominate-by-normalized-blowing-up}
Let $X$ be a scheme which is Noetherian, Nagata, and has dimension $2$.
Let $f : Y \to X$ be a proper birational morphism.
Then there exists a commutative diagram
$$
\xymatrix{
X_n \ar[r] \ar[d] &
X_{n - 1} \ar[r] &
\ldots \ar[r] &
X_1 \ar[r] &
X_0 \ar[d] \\
Y \ar[rrrr] & & & & X
}
$$
where $X_0 \to X$ is the normalization and
where $X_{i + 1} \to X_i$ is the normalized blowing up of $X_i$ at a closed
point.
\end{lemma}
\begin{proof}
We will use the results of Morphisms, Sections
\ref{morphisms-section-nagata},
\ref{morphisms-section-dimension-formula}, and
\ref{morphisms-section-normalization} without further mention.
We may replace $Y$ by its normalization. Let $X_0 \to X$
be the normalization. The morphism $Y \to X$ factors through $X_0$.
Thus we may assume that both $X$ and $Y$ are normal.
\medskip\noindent
Assume $X$ and $Y$ are normal. The morphism $f : Y \to X$ is an isomorphism
over an open which contains every point of codimension $0$ and $1$ in $Y$ and
every point of $Y$ over which the fibre is finite, see Varieties, Lemma
\ref{varieties-lemma-modification-normal-iso-over-codimension-1}.
Hence there is a finite set of closed points $T \subset X$
such that $f$ is an isomorphism over $X \setminus T$. For each $x \in T$
the fibre $Y_x$ is a proper geometrically connected scheme of dimension $1$
over $\kappa(x)$, see
More on Morphisms, Lemma
\ref{more-morphisms-lemma-geometrically-connected-fibres-towards-normal}.
Thus
$$
BadCurves(f) = \{C \subset Y\text{ closed} \mid
\dim(C) = 1, f(C) = \text{a point}\}
$$
is a finite set. We will prove the lemma by induction on the number
of elements of $BadCurves(f)$. The base case is the case where $BadCurves(f)$
is empty, and in that case $f$ is an isomorphism.
\medskip\noindent
Fix $x \in T$. Let $X' \to X$ be the normalized blowup of $X$ at $x$ and let
$Y'$ be the normalization of $Y \times_X X'$. Picture
$$
\xymatrix{
Y' \ar[r]_{f'} \ar[d] & X' \ar[d] \\
Y \ar[r]^f & X
}
$$
Let $x' \in X'$ be a closed point lying over $x$ such that
the fibre $Y'_{x'}$ has dimension $\geq 1$. Let $C' \subset Y'$
be an irreducible component of $Y'_{x'}$, i.e., $C' \in BadCurves(f')$.
Since $Y' \to Y \times_X X'$ is finite we see that $C'$ must map
to an irreducible component $C \subset Y_x$.
If is clear that $C \in BadCurves(f)$.
Since $Y' \to Y$ is birational and hence an isomorphism over points of
codimension $1$ in $Y$, we see that we obtain an injective map
$$
BadCurves(f') \longrightarrow BadCurves(f)
$$
Thus it suffices to show that after a finite number of these
normalized blowups we get rid at of at least one of the bad
curves, i.e., the displayed map is not surjective.
\medskip\noindent
We will get rid of a bad curve using an argument due to Zariski.
Pick $C \in BadCurves(f)$ lying over our $x$. Denote $\mathcal{O}_{Y, C}$
the local ring of $Y$ at the generic point of $C$. Choose an element
$u \in \mathcal{O}_{X, C}$ whose image in the residue field
$R(C)$ is transcendental over $\kappa(x)$ (we can do this because
$R(C)$ has transcendence degree $1$ over $\kappa(x)$ by
Varieties, Lemma \ref{varieties-lemma-dimension-locally-algebraic}).
We can write $u = a/b$ with $a, b \in \mathcal{O}_{X, x}$ as
$\mathcal{O}_{Y, C}$ and $\mathcal{O}_{X, x}$ have the same
fraction fields. By our choice of $u$ it must be the case that
$a, b \in \mathfrak m_x$. Hence
$$
N_{u, a, b} = \min
\{\text{ord}_{\mathcal{O}_{Y, C}}(a), \text{ord}_{\mathcal{O}_{Y, C}}(b)\} > 0
$$
Thus we can do descending induction on this integer.
Let $X' \to X$ be the normalized blowing up of $x$
and let $Y'$ be the normalization of $X' \times_X Y$ as above. We will
show that if $C$ is the image of some bad curve $C' \subset Y'$
lying over $x' \in X'$, then
there exists a choice of $a', b' \mathcal{O}_{X', x'}$
such that $N_{u, a', b'} < N_{u, a, b}$. This will finish the proof.
Namely, since $X' \to X$ factors through the blowing up, we see that
there exists a nonzero element $d \in \mathfrak m_{x'}$ such that
$a = a' d$ and $b = b' d$ (namely, take $d$ to be the local equation
for the exceptional divisor of the blow up). Since $Y' \to Y$
is an isomorphism over an open containing the generic point of $C$
(seen above) we see that $\mathcal{O}_{Y', C'} = \mathcal{O}_{Y, C}$.
Hence
$$
\text{ord}_{\mathcal{O}_{Y, C}}(a) =
\text{ord}_{\mathcal{O}_{Y', C'}}(a' d) =
\text{ord}_{\mathcal{O}_{Y', C'}}(a') +
\text{ord}_{\mathcal{O}_{Y', C'}}(d) >
\text{ord}_{\mathcal{O}_{Y', C'}}(a')
$$
Similarly for $b$ and the proof is complete.
\end{proof}
\begin{lemma}
\label{lemma-extend-rational-map-normalized-blowing-up}
Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is
Noetherian, Nagata, and has dimension $2$. Let $Y$ be a proper
scheme over $S$. Given an $S$-rational map $f : U \to Y$ from
$X$ to $Y$ there exists a sequence
$$
X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 \to X
$$
and an $S$-morphism $f_n : X_n \to Y$ such that $X_0 \to X$ is the
normalization, $X_{i + 1} \to X_i$ is the normalized blowing up of
$X_i$ at a closed point, and $f_n$ and $f$ agree.
\end{lemma}
\begin{proof}
Applying
Divisors, Lemma \ref{divisors-lemma-extend-rational-map-after-modification}
we find a proper morphism $p : X' \to X$ which is an isomorphism
over $U$ and a morphism $f' : X' \to Y$ agreeing with $f$ over $U$.
Apply Lemma \ref{lemma-dominate-by-normalized-blowing-up}
to the morphism $p : X' \to X$. The composition $X_n \to X' \to Y$ is
the desired morphism.
\end{proof}
\section{Modifying over local rings}
\label{section-modifications}
\noindent
Let $S$ be a scheme. Let $s_1, \ldots, s_n \in S$ be pairwise distinct
closed points. Assume that the open embedding
$$
U = S \setminus \{s_1, \ldots, s_n\} \longrightarrow S
$$
is quasi-compact. Denote $FP_{S, \{s_1, \ldots, s_n\}}$
the category of morphisms $f : X \to S$ of finite presentation
which induce an isomorphism $f^{-1}(U) \to U$.
Morphisms are morphisms of schemes over $S$.
For each $i$ set $S_i = \Spec(\mathcal{O}_{S, s_i})$
and let $V_i = S_i \setminus \{s_i\}$. Denote
$FP_{S_i, s_i}$ the category of
morphisms $g_i : Y_i \to S_i$ of finite presentation which induce an
isomorphism $g_i^{-1}(V_i) \to V_i$.
Morphisms are morphisms over $S_i$.
Base change defines an functor
\begin{equation}
\label{equation-equivalence}
F :
FP_{S, \{s_1, \ldots, s_n\}}
\longrightarrow
FP_{S_1, s_1} \times \ldots \times FP_{S_n, s_n}
\end{equation}
To reduce at least some of the problems in this chapter to the case
of local rings we have the following lemma.
\begin{lemma}
\label{lemma-equivalence}
The functor $F$ (\ref{equation-equivalence}) is an equivalence.
\end{lemma}
\begin{proof}
For $n = 1$ this is Limits, Lemma \ref{limits-lemma-modifications}.
For $n > 1$ the lemma can be proved in exactly the same way or it
can be deduced from it. For example, suppose that
$g_i : Y_i \to S_i$ are objects of $\mathcal{C}_{S_i, S_i}$.
Then by the case $n = 1$ we can find $f'_i : X'_i \to S$
of finite presentation
which are isomorphisms over $S \setminus \{s_i\}$ and whose
base change to $S_i$ is $g_i$. Then we can set
$$
f : X = X'_1 \times_S \ldots \times_S X'_n \to S
$$
This is an object of $\mathcal{C}_{S, \{s_1, \ldots, s_n\}}$
whose base change by $S_i \to S$ recovers $g_i$. Thus the functor
is essentially surjective. We omit the proof of
fully faithfulness.
\end{proof}
\begin{lemma}
\label{lemma-equivalence-properties}
Let $S, s_i, S_i$ be as in (\ref{equation-equivalence}).
If $f : X \to S$ corresponds to $g_i : Y_i \to S_i$ under $F$,
then $f$ is separated, proper, finite, if and only if $g_i$ is so
for $i = 1, \ldots, n$.
\end{lemma}
\begin{proof}
Follows from Limits, Lemma
\ref{limits-lemma-modifications-properties}.
\end{proof}
\begin{lemma}
\label{lemma-equivalence-fibre}
Let $S, s_i, S_i$ be as in (\ref{equation-equivalence}).
If $f : X \to S$ corresponds to $g_i : Y_i \to S_i$ under $F$,
then $X_{s_i} \cong (Y_i)_{s_i}$ as schemes over $\kappa(s_i)$.
\end{lemma}
\begin{proof}
This is clear.
\end{proof}
\begin{lemma}
\label{lemma-equivalence-sequence-blowups}
Let $S, s_i, S_i$ be as in (\ref{equation-equivalence})
and assume $f : X \to S$ corresponds to $g_i : Y_i \to S_i$ under $F$.
Then there exists a factorization
$$
X = Z_m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S
$$
of $f$ where $Z_{j + 1} \to Z_j$ is the blowing up of $Z_j$ at a closed
point $z_j$ lying over $\{s_1, \ldots, s_n\}$ if and only if for each
$i$ there exists a factorization
$$
Y_i = Z_{i, m_i} \to Z_{i, m_i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_i
$$
of $g_i$ where $Z_{i, j + 1} \to Z_{i, j}$ is the blowing up of $Z_{i, j}$
at a closed point $z_{i, j}$ lying over $s_i$.
\end{lemma}
\begin{proof}
Let's start with a sequence of blowups
$Z_m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S$.
The first morphism $Z_1 \to S$ is given
by blowing up one of the $s_i$, say $s_1$. Applying $F$
to $Z_1 \to S$ we find a blow up $Z_{1, 1} \to S_1$ at $s_1$
is the blowing up at $s_1$ and otherwise $Z_{i, 0} = S_i$ for $i > 1$.
In the next step, we either blow up one of the $s_i$, $i \geq 2$
on $Z_1$ or we pick a closed point $z_1$ of the fibre of $Z_1 \to S$
over $s_1$. In the first case it is clear what to do and in
the second case we use that $(Z_1)_{s_1} \cong (Z_{1, 1})_{s_1}$
(Lemma \ref{lemma-equivalence-fibre})
to get a closed point $z_{1, 1} \in Z_{1, 1}$ corresponding to $z_1$.
Then we set $Z_{1, 2} \to Z_{1, 1}$ equal to the blowing up
in $z_{1, 1}$. Continuing in this manner we construct the factorizations
of each $g_i$.
\medskip\noindent
Conversely, given sequences of blowups
$Z_{i, m_i} \to Z_{i, m_i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_i$
we construct the sequence of blowing ups of $S$ in exactly the same manner.
\end{proof}
\noindent
Here is the analogue of
Lemma \ref{lemma-equivalence-sequence-blowups}
for normalized blowups.
\begin{lemma}
\label{lemma-equivalence-sequence-normalized-blowups}
Let $S, s_i, S_i$ be as in (\ref{equation-equivalence})
and assume $f : X \to S$ corresponds to $g_i : Y_i \to S_i$ under $F$.
Assume every quasi-compact open of $S$ has finitely many irreducible
components. Then there exists a factorization
$$
X = Z_m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S
$$
of $f$ where $Z_{j + 1} \to Z_j$ is the normalized blowing up of $Z_j$
at a closed point $z_j$ lying over $\{x_1, \ldots, x_n\}$ if and only if
for each $i$ there exists a factorization
$$
Y_i = Z_{i, m_i} \to Z_{i, m_i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_i
$$
of $g_i$ where $Z_{i, j + 1} \to Z_{i, j}$ is the normalized blowing up of
$Z_{i, j}$ at a closed point $z_{i, j}$ lying over $s_i$.
\end{lemma}
\begin{proof}
The assumption on $S$ is used to assure us (successively) that
the schemes we are normalizing have locally finitely many irreducible
components so that the statement makes sense. Having said this the
lemma follows by the exact same argument as used to prove
Lemma \ref{lemma-equivalence-sequence-blowups}.
\end{proof}
\section{Vanishing}
\label{section-vanishing}
\noindent
In this section we will often work in the following setting.
Recall that a modification is a proper birational morphism
between integral schemes (Morphisms, Definition
\ref{morphisms-definition-modification}).
\begin{situation}
\label{situation-vanishing}
Here $(A, \mathfrak m, \kappa)$ be a local Noetherian normal domain of
dimension $2$. Let $s$ be the closed point of $S = \Spec(A)$ and
$U = S \setminus \{s\}$. Let $f : X \to S$ be a modification.
We denote $C_1, \ldots, C_r$ the irreducible
components of the special fibre $X_s$ of $f$.
\end{situation}
\noindent
By Varieties, Lemma
\ref{varieties-lemma-modification-normal-iso-over-codimension-1}
the morphism $f$ defines an isomorphism $f^{-1}(U) \to U$.
The special fibre $X_s$ is proper over $\Spec(\kappa)$ and
has dimension at most $1$ by Varieties, Lemma
\ref{varieties-lemma-dimension-fibre-in-higher-codimension}.
By Stein factorization (More on Morphisms, Lemma
\ref{more-morphisms-lemma-geometrically-connected-fibres-towards-normal})
we have $f_*\mathcal{O}_X = \mathcal{O}_S$ and
the special fibre $X_s$ is geometrically connected over $\kappa$.
If $X_s$ has dimension $0$, then $f$ is finite
(More on Morphisms, Lemma
\ref{more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood})
and hence an isomorphism
(Morphisms, Lemma \ref{morphisms-lemma-finite-birational-over-normal}).
We will discard this uninteresting case and we conclude that
$\dim(C_i) = 1$ for $i = 1, \ldots, r$.
\begin{lemma}
\label{lemma-dominate-by-scheme-modification}
In Situation \ref{situation-vanishing} there exists a $U$-admissible
blowup $X' \to S$ which dominates $X$.
\end{lemma}
\begin{proof}
This is a special case of
More on Flatness, Lemma \ref{flat-lemma-dominate-modification-by-blowup}.
\end{proof}
\begin{lemma}
\label{lemma-nice-meromorphic-function}
In Situation \ref{situation-vanishing} there exists a nonzero
$f \in \mathfrak m$ such that for every $i = 1, \ldots, r$ there exist
\begin{enumerate}
\item a closed point $x_i \in C_i$ with $x_i \not \in C_j$ for $j \not = i$,
\item a factorization $f = g_i f_i$ of $f$ in $\mathcal{O}_{X, x_i}$
such that $g_i \in \mathfrak m_{x_i}$ maps to a nonzero element
of $\mathcal{O}_{C_i, x_i}$.
\end{enumerate}
\end{lemma}
\begin{proof}
We will use the observations made following Situation \ref{situation-vanishing}
without further mention. Pick a closed point $x_i \in C_i$ which is not in
$C_j$ for $j \not = i$. Pick $g_i \in \mathfrak m_{x_i}$ which maps to a
nonzero element of $\mathcal{O}_{C_i, x_i}$. Since the fraction field of $A$
is the fraction field of $\mathcal{O}_{X_i, x_i}$ we can write
$g_i = a_i/b_i$ for some $a_i, b_i \in A$. Take $f = \prod a_i$.
\end{proof}
\begin{lemma}
\label{lemma-nontrivial-normal-bundle}
In Situation \ref{situation-vanishing} assume $X$ is normal.
Let $Z \subset X$ be a nonempty effective Cartier divisor such that
$Z \subset X_s$ set theoretically.
Then the conormal sheaf of $Z$ is not trivial.
More precisely, there exists an $i$ such that $C_i \subset Z$
and $\deg(\mathcal{C}_{Z/X}|_{C_i}) > 0$.
\end{lemma}
\begin{proof}
We will use the observations made following Situation \ref{situation-vanishing}
without further mention. Let $f$ be a function as in
Lemma \ref{lemma-nice-meromorphic-function}.
Let $\xi_i \in C_i$ be the generic point. Let
$\mathcal{O}_i$ be the local ring of $X$ at $\xi_i$. Then $\mathcal{O}_i$
is a discrete valuation ring. Let $e_i$ be the valuation of
$f$ in $\mathcal{O}_i$, so $e_i > 0$. Let $h_i \in \mathcal{O}_i$ be a local
equation for $Z$ and let $d_i$ be its valuation. Then $d_i \geq 0$.
Choose and fix $i$ with $d_i/e_i$ maximal (then $d_i > 0$ as
$Z$ is not empty). Replace $f$ by $f^{d_i}$ and $Z$ by $e_iZ$.
This is permissible, by the relation
$\mathcal{O}_X(e_i Z) = \mathcal{O}_X(Z)^{\otimes e_i}$,
the relation between the conormal sheaf and $\mathcal{O}_X(Z)$
(see Divisors, Lemmas
\ref{divisors-lemma-invertible-sheaf-sum-effective-Cartier-divisors}
and \ref{divisors-lemma-conormal-effective-Cartier-divisor}, and
since the degree gets multiplied by $e_i$, see
Varieties, Lemma \ref{varieties-lemma-degree-tensor-product}.
Let $\mathcal{I}$ be the ideal sheaf of $Z$ so that
$\mathcal{C}_{Z/X} = \mathcal{I}|_Z$. Consider the image $\overline{f}$
of $f$ in $\Gamma(Z, \mathcal{O}_Z)$. By our choices above we see
that $\overline{f}$ vanishes in the generic points of irreducible
components of $Z$ (these are all generic points of $C_j$ as $Z$ is
contained in the special fibre). On the other hand, $Z$ is $(S_1)$ by
Divisors, Lemma \ref{divisors-lemma-normal-effective-Cartier-divisor-S1}.
Thus the scheme $Z$ has no embedded associated points and
we conclude that $\overline{f} = 0$ (Divisors, Lemmas
\ref{divisors-lemma-S1-no-embedded} and
\ref{divisors-lemma-restriction-injective-open-contains-weakly-ass}).
Hence $f$ is a global section of $\mathcal{I}$
which generates $\mathcal{I}_{\xi_i}$ by construction.
Thus the image $s_i$ of $f$ in $\Gamma(C_i, \mathcal{I}|_{C_i})$ is nonzero.
However, our choice of $f$ guarantees that $s_i$ has a zero at $x_i$.
Hence the degree of $\mathcal{I}|_{C_i}$ is $>0$ by
Varieties, Lemma \ref{varieties-lemma-check-invertible-sheaf-trivial}.
\end{proof}
\begin{lemma}
\label{lemma-H1-injective}
In Situation \ref{situation-vanishing} assume $X$ is normal
and $A$ Nagata. The map
$$
H^1(X, \mathcal{O}_X) \longrightarrow H^1(f^{-1}(U), \mathcal{O}_X)
$$
is injective.
\end{lemma}
\begin{proof}
Let $0 \to \mathcal{O}_X \to \mathcal{E} \to \mathcal{O}_X \to 0$ be the
extension corresponding to a nontrivial element $\xi$ of
$H^1(X, \mathcal{O}_X)$
(Cohomology, Lemma \ref{cohomology-lemma-h1-extensions}).
Let $\pi : P = \mathbf{P}(\mathcal{E}) \to X$
be the projective bundle associated to $\mathcal{E}$.
The surjection $\mathcal{E} \to \mathcal{O}_X$
defines a section $\sigma : X \to P$ whose conormal sheaf is
isomorphic to $\mathcal{O}_X$ (Divisors, Lemma
\ref{divisors-lemma-conormal-sheaf-section-projective-bundle}).
If the restriction of $\xi$ to $f^{-1}(U)$ is trivial, then we get
a map $\mathcal{E}|_{f^{-1}(U)} \to \mathcal{O}_{f^{-1}(U)}$ splitting
the injection $\mathcal{O}_X \to \mathcal{E}$. This defines a second
section $\sigma' : f^{-1}(U) \to P$ disjoint from $\sigma$. Since $\xi$
is nontrivial we conclude that $\sigma'$ cannot extend to all of $X$
and be disjoint from $\sigma$. Let $X' \subset P$ be the
scheme theoretic image of $\sigma'$ (Morphisms,
Definition \ref{morphisms-definition-scheme-theoretic-image}).
Picture
$$
\xymatrix{
& X' \ar[rd]_g \ar[r] & P \ar[d]_\pi \\
f^{-1}(U) \ar[ru]_{\sigma'} \ar[rr] & & X \ar@/_/[u]_\sigma
}
$$
The morphism $P \setminus \sigma(X) \to X$ is affine.
If $X' \cap \sigma(X) = \emptyset$, then $X' \to X$ is both affine
and proper, hence finite
(Morphisms, Lemma \ref{morphisms-lemma-finite-proper}),
hence an isomorphism (as $X$ is normal, see
Morphisms, Lemma \ref{morphisms-lemma-finite-birational-over-normal}).
This is impossible as mentioned above.
\medskip\noindent
Let $X^\nu$ be the normalization of $X'$.
Since $A$ is Nagata, we see that $X^\nu \to X'$ is finite
(Morphisms, Lemmas \ref{morphisms-lemma-nagata-normalization} and
\ref{morphisms-lemma-ubiquity-nagata}). Let $Z \subset X^\nu$ be the
pullback of the effective Cartier divisor $\sigma(X) \subset P$.
By the above we see that $Z$ is not empty and is contained
in the closed fibre of $X^\nu \to S$.
Since $P \to X$ is smooth, we see that $\sigma(X)$ is an effective
Cartier divisor
(Divisors, Lemma \ref{divisors-lemma-section-smooth-regular-immersion}).
Hence $Z \subset X^\nu$ is an effective Cartier divisor too.
Since the conormal sheaf of $\sigma(X)$ in $P$ is $\mathcal{O}_X$, the
conormal sheaf of $Z$ in $X^\nu$ (which is a priori invertible)
is $\mathcal{O}_Z$ by
Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial-flat}.
This is impossible by
Lemma \ref{lemma-nontrivial-normal-bundle}
and the proof is complete.
\end{proof}
\begin{lemma}
\label{lemma-R1-injective}
In Situation \ref{situation-vanishing} assume $X$ is normal and $A$ Nagata.
Then
$$
\Hom_{D(A)}(\kappa[-1], Rf_*\mathcal{O}_X)
$$
is zero. This uses $D(A) = D_\QCoh(\mathcal{O}_S)$ to think of
$Rf_*\mathcal{O}_X$ as an object of $D(A)$.
\end{lemma}
\begin{proof}
By adjointness of $Rf_*$ and $Lf^*$ such a map is the same thing
as a map $\alpha : Lf^*\kappa[-1] \to \mathcal{O}_X$. Note that
$$
H^i(Lf^*\kappa[-1]) =
\left\{
\begin{matrix}
0 & \text{if} & i > 1 \\
\mathcal{O}_{X_s} & \text{if} & i = 1 \\
\text{some }\mathcal{O}_{X_s}\text{-module} & \text{if} & i \leq 0
\end{matrix}
\right.
$$
Since $\Hom(H^0(Lf^*\kappa[-1]), \mathcal{O}_X) = 0$ as $\mathcal{O}_X$
is torsion free, the spectral sequence for $\text{Ext}$
(Cohomology on Sites, Example
\ref{sites-cohomology-example-hom-complex-into-sheaf})
implies that
$\Hom_{D(\mathcal{O}_X)}(Lf^*\kappa[-1], \mathcal{O}_X)$ is equal to
$\text{Ext}^1_{\mathcal{O}_X}(\mathcal{O}_{X_s}, \mathcal{O}_X)$.
We conclude that
$\alpha : Lf^*\kappa[-1] \to \mathcal{O}_X$ is given by an extension
$$
0 \to \mathcal{O}_X \to \mathcal{E} \to \mathcal{O}_{X_s} \to 0
$$
By Lemma \ref{lemma-H1-injective} the pullback of this extension
via the surjection $\mathcal{O}_X \to \mathcal{O}_{X_s}$ is zero
(since this pullback is clearly split over $f^{-1}(U)$).
Thus $1 \in \mathcal{O}_{X_s}$ lifts to a global section $s$ of
$\mathcal{E}$. Multiplying $s$ by the ideal sheaf $\mathcal{I}$
of $X_s$ we obtain an $\mathcal{O}_X$-module map
$c_s : \mathcal{I} \to \mathcal{O}_X$. Applying $f_*$ we obtain
an $A$-linear map $f_*c_s : \mathfrak m \to A$. Since $A$ is
a Noetherian normal local domain this map is given by multiplication
by an element $a \in A$. Changing $s$ into $s - a$ we find that
$s$ is annihilated by $\mathcal{I}$ and the extension is trivial
as desired.
\end{proof}
\begin{remark}
\label{remark-dualizing-setup}
Let $X$ be an integral Noetherian normal scheme of dimension $2$.
In this case the following are equivalent
\begin{enumerate}
\item $X$ has a dualizing complex $\omega_X^\bullet$,
\item there is a coherent $\mathcal{O}_X$-module $\omega_X$ such that
$\omega_X[n]$ is a dualizing complex, where $n$ can be any integer.
\end{enumerate}
This follows from the fact that $X$ is Cohen-Macaulay
(Properties, Lemma \ref{properties-lemma-normal-dimension-2-Cohen-Macaulay}) and
Dualizing Complexes, Lemma \ref{dualizing-lemma-dualizing-module-CM-scheme}.
In this situation we will say that $\omega_X$ is a {\it dualizing module}
in accordance with
Dualizing Complexes, Section \ref{dualizing-section-dualizing-module}.
In particular, when $A$ is a Noetherian normal local domain of dimension
$2$, then we say {\it $A$ has a dualizing module $\omega_A$}
if the above is true. In this case, if $X \to \Spec(A)$ is a normal
modification, then $X$ has a dualizing module too, see
Dualizing Complexes, Example \ref{dualizing-example-proper-over-local}.
In this situation we always denote $\omega_X$ the dualizing
module normalized with respect to $\omega_A$, i.e., such that
$\omega_X[2]$ is the dualizing complex normalized relative to
$\omega_A[2]$. See Dualizing Complexes, Section \ref{dualizing-section-glue}.
\end{remark}
\noindent
The Grauert-Riemenschneider vanishing of the next proposition is a formal
consequence of Lemma \ref{lemma-R1-injective} and the general theory of
duality.
\begin{proposition}[Grauert-Riemenschneider]
\label{proposition-Grauert-Riemenschneider}
In Situation \ref{situation-vanishing} assume
\begin{enumerate}
\item $X$ is a normal scheme,
\item $A$ is Nagata and has a dualizing complex $\omega_A^\bullet$.
\end{enumerate}
Let $\omega_X$ be the dualizing module of $X$
(Remark \ref{remark-dualizing-setup}). Then $R^1f_*\omega_X = 0$.
\end{proposition}
\begin{proof}
In this proof we will use the identification $D(A) = D_\QCoh(\mathcal{O}_S)$
to identify quasi-coherent $\mathcal{O}_S$-modules with $A$-modules.
Moreover, we may assume that $\omega_A^\bullet$ is normalized, see
Dualizing Complexes, Section \ref{dualizing-section-dualizing-local}.
Since $X$ is a Noetherian normal $2$-dimensional scheme
it is Cohen-Macaulay (Properties, Lemma
\ref{properties-lemma-normal-dimension-2-Cohen-Macaulay}).
Thus $\omega_X^\bullet = \omega_X[2]$ (Dualizing Complexes, Lemma
\ref{dualizing-lemma-dualizing-module-CM-scheme} and the
normalization in Dualizing Complexes, Example
\ref{dualizing-example-proper-over-local}).
If the proposition is false, then we can find a nonzero map
$R^1f_*\omega_X \to \kappa$. In other words we obtain a nonzero map
$\alpha : Rf_*\omega_X^\bullet \to \kappa[1]$.
Applying $R\Hom_A(-, \omega_A^\bullet)$ we get a nonzero map
$$
\beta : \kappa[-1] \longrightarrow Rf_*\mathcal{O}_X
$$
which is impossible by Lemma \ref{lemma-R1-injective}.
To see that $R\Hom_A(-, \omega_A^\bullet)$ does what we said, first
note that
$$
R\Hom_A(\kappa[1], \omega_A^\bullet) =
R\Hom_A(\kappa, \omega_A^\bullet)[-1] =
\kappa[-1]
$$
as $\omega_A^\bullet$ is normalized and we have
$$
R\Hom_A(Rf_*\omega_X^\bullet, \omega_A^\bullet) =
Rf_*R\SheafHom_{\mathcal{O}_X}(\omega_X^\bullet, \omega_X^\bullet) =
Rf_*\mathcal{O}_X
$$
The first equality by
Dualizing Complexes, Lemma \ref{dualizing-lemma-proper-noetherian-relative}
and the fact that $\omega_X^\bullet = f^!\omega_A^\bullet$
by construction, and the second equality because $\omega_X^\bullet$
is a dualizing complex for $X$ (which goes back to
Dualizing Complexes, Lemma \ref{dualizing-lemma-shriek-dualizing}).
\end{proof}
\section{Boundedness}
\label{section-bounded}
\noindent
In this section we begin the discussion which will lead to a reduction to
the case of rational singularities for $2$-dimensional schemes.
\begin{lemma}
\label{lemma-exact-sequence}
Let $(A, \mathfrak m, \kappa)$ be a Noetherian normal local domain
of dimension $2$. Consider a commutative diagram
$$
\xymatrix{
X' \ar[rd]_{f'} \ar[rr]_g & & X \ar[ld]^f \\
& \Spec(A)
}
$$
where $f$ and $f'$ are modifications as in Situation \ref{situation-vanishing}
and $X$ normal. Then we have a short exact sequence
$$
0 \to H^1(X, \mathcal{O}_X) \to H^1(X', \mathcal{O}_{X'}) \to
H^0(X, R^1g_*\mathcal{O}_{X'}) \to 0
$$
Also $\dim(\text{Supp}(R^1g_*\mathcal{O}_{X'})) = 0$
and $R^1g_*\mathcal{O}_{X'}$ is generated by global sections.
\end{lemma}
\begin{proof}
We will use the observations made following Situation \ref{situation-vanishing}
without further mention. As $X$ is normal and $g$ is dominant and
birational, we have $g_*\mathcal{O}_{X'} = \mathcal{O}_X$, see for
example More on Morphisms, Lemma
\ref{more-morphisms-lemma-geometrically-connected-fibres-towards-normal}.
Since the fibres of $g$ have dimension $\leq 1$, we have
$R^pg_*\mathcal{O}_{X'} = 0$ for $p > 1$, see for example
Cohomology of Schemes, Lemma
\ref{coherent-lemma-higher-direct-images-zero-above-dimension-fibre}.
The support of $R^1g_*\mathcal{O}_{X'}$ is contained in the set of points of
$X$ where the fibres of $g'$ have dimension $\geq 1$. Thus
it is contained in the set of images of those
irreducible components $C' \subset X'_s$ which map to points of $X_s$
which is a finite set of closed points
(recall that $X'_s \to X_s$ is a morphism of proper $1$-dimensional
schemes over $\kappa$). Then $R^1g_*\mathcal{O}_{X'}$ is globally
generated by
Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-support-dimension-0}.
Using the morphism $f : X \to S$ and the references above we find that
$H^p(X, \mathcal{F}) = 0$ for $p > 1$ for any coherent $\mathcal{O}_X$-module
$\mathcal{F}$. Hence the short exact sequence of the lemma is a consequence
of the Leray spectral sequence for $g$ and $\mathcal{O}_{X'}$, see
Cohomology, Lemma \ref{cohomology-lemma-Leray}.
\end{proof}
\begin{lemma}
\label{lemma-bound-a-torsion}
Let $(A, \mathfrak m, \kappa)$ be a local normal Nagata domain
of dimension $2$. Let $a \in A$ be nonzero. There exists an integer $N$ such
that for every modification $f : X \to \Spec(A)$ with $X$ normal the
$A$-module
$$
M_{X, a} = \Coker(A \longrightarrow H^0(Z, \mathcal{O}_Z))
$$
where $Z \subset X$ is cut out by $a$ has length bounded by $N$.
\end{lemma}
\begin{proof}
By the short exact sequence
$
0 \to \mathcal{O}_X \xrightarrow{a} \mathcal{O}_X \to \mathcal{O}_Z \to 0
$
we see that
\begin{equation}
\label{equation-a-torsion}
M_{X, a} = H^1(X, \mathcal{O}_X)[a]
\end{equation}
Here $N[a] = \{n \in N \mid an = 0\}$ for an $A$-module $N$. Thus
if $a$ divides $b$, then $M_{X, a} \subset M_{X, b}$.
Suppose that for some $c \in A$ the modules $M_{X, c}$
have bounded length. Then for every $X$ we have an exact sequence
$$
0 \to M_{X, c} \to M_{X, c^2} \to M_{X, c}
$$
where the second arrow is given by multiplication by $c$. Hence we see that
$M_{X, c^2}$ has bounded length as well. Thus it suffices to find a $c \in A$
for which the lemma is true such that $a$ divides $c^n$ for some $n > 0$.
By More on Algebra, Lemma \ref{more-algebra-lemma-divides-radical}
we may assume $A/(a)$ is a reduced ring.
\medskip\noindent
Assume that $A/(a)$ is reduced. Let $A/(a) \subset B$ be the normalization
of $A/(a)$ in its quotient ring. Because $A$ is Nagata, we see that
$\text{Coker}(A \to B)$ is finite. We claim the length of this finite
module is a bound. To see this, consider $f : X \to \Spec(A)$ as in the lemma
and let $Z' \subset Z$ be the scheme theoretic closure of $Z \cap f^{-1}(U)$.
Then $Z' \to \Spec(A/(a))$ is finite for example by Varieties, Lemma
\ref{varieties-lemma-finite-in-codim-1}.
Hence $Z' = \Spec(B')$ with $A/(a) \subset B' \subset B$.
On the other hand, we claim the map
$$
H^0(Z, \mathcal{O}_Z) \to H^0(Z', \mathcal{O}_{Z'})
$$
is injective. Namely, if $s \in H^0(Z, \mathcal{O}_Z)$
is in the kernel, then
the restriction of $s$ to $f^{-1}(U) \cap Z$ is zero.
Hence the image of $s$ in $H^1(X, \mathcal{O}_X)$ vanishes in
$H^1(f^{-1}(U), \mathcal{O}_X)$. By Lemma \ref{lemma-H1-injective}
we see that $s$ comes from an element $\tilde s$ of $A$. But by
assumption $\tilde s$ maps to zero in $B'$ which implies that $s = 0$.
Putting everything together we see that
$M_{X, a}$ is a subquotient of $B'/A$, namely not every element
of $B'$ extends to a global section of $\mathcal{O}_Z$, but in
any case the length of $M_{X, a}$ is bounded by the length of $B/A$.
\end{proof}
\noindent
In some cases, resolution of singularities reduces to the case
of rational singularities.
\begin{definition}
\label{definition-reduce-to-rational}
Let $(A, \mathfrak m, \kappa)$ be a local normal Nagata domain
of dimension $2$.
\begin{enumerate}
\item We say $A$ {\it defines a rational singularity} if for every
normal modification $X \to \Spec(A)$ we have $H^1(X, \mathcal{O}_X) = 0$.
\item We say that {\it reduction to rational singularities
is possible for $A$} if the length of the $A$-modules
$$
H^1(X, \mathcal{O}_X)
$$
is bounded for all modifications $X \to \Spec(A)$ with $X$ normal.
\end{enumerate}
\end{definition}
\noindent
The meaning of the language in (2) is explained by
Lemma \ref{lemma-reduce-to-rational}. The following lemma
says roughly speaking that local rings of modifications of $\Spec(A)$
with $A$ defining a rational singularity also define rational
singularities.
\begin{lemma}
\label{lemma-rational-propagates}
Let $(A, \mathfrak m, \kappa)$ be a local normal Nagata domain of
dimension $2$ which defines a rational singularity. Let $A \subset B$
be a local extension of domains with the same fraction field
which is essentially of finite type such
that $\dim(B) = 2$ and $B$ normal. Then $B$ defines a rational singularity.
\end{lemma}
\begin{proof}
Choose a finite type $A$-algebra $C$ such that $B = C_\mathfrak q$
for some prime $\mathfrak q \subset C$. After replacing
$C$ by the image of $C$ in $B$ we may assume that $C$ is a domain
with fraction field equal to the fraction field of $A$.
Then we can choose a closed immersion $\Spec(C) \to \mathbf{A}^n_A$
and take the closure in $\mathbf{P}^n_A$ to conclude that $B$
is isomorphic to $\mathcal{O}_{X, x}$ for some closed point $x \in X$
of a projective modification $X \to \Spec(A)$.
(Morphisms, Lemma \ref{morphisms-lemma-dimension-formula},
shows that $\kappa(x)$ is finite over $\kappa$ and then
Morphisms, Lemma
\ref{morphisms-lemma-algebraic-residue-field-extension-closed-point-fibre}
shows that $x$ is a closed point.)
Let $\nu : X^\nu \to X$ be the normalization.
Since $A$ is Nagata the morphism $\nu$ is finite (Morphisms, Lemma
\ref{morphisms-lemma-nagata-normalization}).
Thus $X^\nu$ is projective over $A$ by
More on Morphisms, Lemma
\ref{more-morphisms-lemma-category-projective}.
Since $B = \mathcal{O}_{X, x}$ is normal, we see that
$\mathcal{O}_{X, x} = (\nu_*\mathcal{O}_{X^\nu})_x$.
Hence there is a unique point $x^\nu \in X^\nu$ lying over $x$
and $\mathcal{O}_{X^\nu, x^\nu} = \mathcal{O}_{X, x}$.
Thus we may assume $X$ is normal and projective over $A$.
Let $Y \to \Spec(\mathcal{O}_{X, x}) = \Spec(B)$
be a modification with $Y$ normal.
We have to show that $H^1(Y, \mathcal{O}_Y) = 0$. By
Limits, Lemma \ref{limits-lemma-modifications}
we can find a morphism of schemes $g : X' \to X$ which is an isomorphism
over $X \setminus \{x\}$ such that $X' \times_X \Spec(\mathcal{O}_{X, x})$
is isomorphic to $Y$. Then $g$ is a modification as it is proper by
Limits, Lemma \ref{limits-lemma-modifications-properties}.
The local ring of $X'$ at a point of $x'$ is either isomorphic
to the local ring of $X$ at $g(x')$ if $g(x') \not = x$ and
if $g(x') = x$, then the local ring of $X'$ at $x'$ is isomorphic
to the local ring of $Y$ at the corresponding point. Hence we see
that $X'$ is normal as both $X$ and $Y$ are normal.
Thus $H^1(X', \mathcal{O}_{X'}) = 0$ by our assumption on $A$.
By Lemma \ref{lemma-exact-sequence} we have $R^1g_*\mathcal{O}_{X'} = 0$.
Clearly this means that $H^1(Y, \mathcal{O}_Y) = 0$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-reduce-to-rational}
Let $(A, \mathfrak m, \kappa)$ be a local normal Nagata domain
of dimension $2$. If reduction to rational singularities is possible
for $A$, then there exists a finite sequence of normalized blowups
$$
X = X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \Spec(A)
$$
in closed points such that for any closed point $x \in X$
the local ring $\mathcal{O}_{X, x}$ defines a rational singularity.
In particular $X \to \Spec(A)$ is a modification and $X$
is a normal scheme projective over $A$.
\end{lemma}
\begin{proof}
We choose a modification $X \to \Spec(A)$ with $X$ normal
which maximizes the length of $H^1(X, \mathcal{O}_X)$.
By Lemma \ref{lemma-exact-sequence}
for any further modification $g : X' \to X$ with $X'$ normal
we have $R^1g_*\mathcal{O}_{X'} = 0$ and
$H^1(X, \mathcal{O}_X) = H^1(X', \mathcal{O}_{X'})$.
\medskip\noindent
Let $x \in X$ be a closed point. We will show that $\mathcal{O}_{X, x}$
defines a rational singularity. Let $Y \to \Spec(\mathcal{O}_{X, x})$
be a modification with $Y$ normal. We have to show that
$H^1(Y, \mathcal{O}_Y) = 0$. By
Limits, Lemma \ref{limits-lemma-modifications}
we can find a morphism of schemes $g : X' \to X$ which is an isomorphism
over $X \setminus \{x\}$ such that $X' \times_X \Spec(\mathcal{O}_{X, x})$
is isomorphic to $Y$. Then $g$ is a modification as it is proper by
Limits, Lemma \ref{limits-lemma-modifications-properties}.
The local ring of $X'$ at a point of $x'$ is either isomorphic
to the local ring of $X$ at $g(x')$ if $g(x') \not = x$ and
if $g(x') = x$, then the local ring of $X'$ at $x'$ is isomorphic
to the local ring of $Y$ at the corresponding point. Hence we see
that $X'$ is normal as both $X$ and $Y$ are normal. By maximality
we have $R^1g_*\mathcal{O}_{X'} = 0$ (see first paragraph). Clearly
this means that $H^1(Y, \mathcal{O}_Y) = 0$ as desired.
\medskip\noindent
The conclusion is that we've found one normal modification $X$
of $\Spec(A)$ such that the local rings of $X$ at closed points all define
rational singularities. Then we choose a sequence of normalized
blowups $X_n \to \ldots \to X_1 \to \Spec(A)$ such that $X_n$
dominates $X$, see Lemma \ref{lemma-dominate-by-normalized-blowing-up}.
For a closed point $x' \in X_n$ mapping to $x \in X$ we can apply
Lemma \ref{lemma-rational-propagates} to the ring map
$\mathcal{O}_{X, x} \to \mathcal{O}_{X_n, x'}$
to see that $\mathcal{O}_{X_n, x'}$ defines a rational singularity.
\end{proof}
\begin{lemma}
\label{lemma-go-up-separable}
Let $A \to B$ be a finite injective local ring map of local normal
Nagata domains of dimension $2$. Assume that the induced extension of
fraction fields is separable. If reduction to rational singularities
is possible for $A$ then it is possible for $B$.
\end{lemma}
\begin{proof}
Let $n$ be the degree of the fraction field extension $K \subset L$.
Let $\text{Trace}_{L/K} : L \to K$ be the trace. Since the extension is finite
separable the trace pairing $(h, g) \mapsto \text{Trace}_{L/K}(fg)$
is a nondegenerate bilinear form on $L$ over $K$. See
Fields, Lemma \ref{fields-lemma-separable-trace-pairing}.
Pick $b_1, \ldots, b_n \in B$ which form a basis of $L$ over $K$.
By the above $d = \det(\text{Trace}_{L/K}(b_ib_j)) \in A$ is nonzero.
\medskip\noindent
Let $Y \to \Spec(B)$ be a modification with $Y$ normal. We can find
a $U$-admissible blow up $X'$ of $\Spec(A)$ such that the strict transform
$Y'$ of $Y$ is finite over $X'$, see More on Flatness, Lemma
\ref{flat-lemma-finite-after-blowing-up}. Picture
$$
\xymatrix{
Y' \ar[d] \ar[r] & Y \ar[r] & \Spec(B) \ar[d] \\
X' \ar[rr] & & \Spec(A)
}
$$
After replacing $X'$ and $Y'$ by their normalizations we may assume that
$X'$ and $Y'$ are normal modifications of $\Spec(A)$ and $\Spec(B)$.
In this way we reduce to the case where there exists a commutative diagram
$$
\xymatrix{
Y \ar[d]_\pi \ar[r]_-g & \Spec(B) \ar[d] \\
X \ar[r]^-f & \Spec(A)
}
$$
with $X$ and $Y$ normal modifications of $\Spec(A)$ and $\Spec(B)$ and
$\pi$ finite.
\medskip\noindent
The trace map on $L$ over $K$ extends to a map of $\mathcal{O}_X$-modules
$\text{Trace} : \pi_*\mathcal{O}_Y \to \mathcal{O}_X$. Consider the map
$$
\Phi : \pi_*\mathcal{O}_Y \longrightarrow \mathcal{O}_X^{\oplus n},\quad
s \longmapsto (\text{Trace}(b_1s), \ldots, \text{Trace}(b_ns))
$$
This map is injective (because it is injective in the generic point)
and there is a map
$$
\mathcal{O}_X^{\oplus n} \longrightarrow \pi_*\mathcal{O}_Y,\quad
(s_1, \ldots, s_n) \longmapsto \sum b_i s_i
$$
whose composition with $\Phi$ has matrix $\text{Trace}(b_ib_j)$.
Hence the cokernel of $\Phi$ is annihilated by $d$. Thus we see that
we have an exact sequence
$$
H^0(X, \Coker(\Phi)) \to H^1(Y, \mathcal{O}_Y) \to
H^1(X, \mathcal{O}_X)^{\oplus n}
$$
Since the right hand side is bounded by assumption, it suffices to show
that the $d$-torsion in $H^1(Y, \mathcal{O}_Y)$ is bounded.
This is the content of Lemma \ref{lemma-bound-a-torsion} and
(\ref{equation-a-torsion}).
\end{proof}
\begin{lemma}
\label{lemma-regular-rational}
Let $A$ be a Nagata regular local ring of dimension $2$. Then $A$ defines
a rational singularity.
\end{lemma}
\begin{proof}
(The assumption that $A$ be Nagata is not necessary for this proof,
but we've only defined the notion of rational singularity in the
case of Nagata $2$-dimensional normal local domains.)
Let $X \to \Spec(A)$ be a modification with $X$ normal. By
Lemma \ref{lemma-dominate-by-blowing-up-in-points}
we can dominate $X$ by a scheme $X_n$ which is the last in a sequence
$$
X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \Spec(A)
$$
of blowing ups in closed points. By Lemma \ref{lemma-blowup-regular}
the schemes $X_i$ are regular, in particular
normal (Algebra, Lemma \ref{algebra-lemma-regular-normal}).
By Lemma \ref{lemma-exact-sequence} we have
$H^1(X, \mathcal{O}_X) \subset H^1(X_n, \mathcal{O}_{X_n})$.
Thus it suffices to prove $H^1(X_n, \mathcal{O}_{X_n}) = 0$.
Using Lemma \ref{lemma-exact-sequence} again, we
see that it suffices to prove $R^1(X_i \to X_{i - 1})_*\mathcal{O}_{X_i} = 0$
for $i = 1, \ldots, n$. This follows from
Lemma \ref{lemma-cohomology-of-blowup}.
\end{proof}
\begin{lemma}
\label{lemma-bound-dualizing-implies-bound}
Let $A$ be a local normal Nagata domain of dimension $2$ which has a
dualizing complex $\omega_A^\bullet$. If there exists a nonzero $d \in A$
such that for all normal modifications $X \to \Spec(A)$ the cokernel of the
trace map
$$
\Gamma(X, \omega_X) \to \omega_A
$$
is annihilated by $d$, then reduction to rational singularities
is possible for $A$.
\end{lemma}
\begin{proof}
For $X \to \Spec(A)$ as in the statement we have to bound
$H^1(X, \mathcal{O}_X)$. Let $\omega_X$ be the dualizing module
of $X$ as in the statement of Grauert-Riemenschneider
(Proposition \ref{proposition-Grauert-Riemenschneider}).
The trace map is the map $Rf_*\omega_X \to \omega_A$ described
in Dualizing Complexes, Section \ref{dualizing-section-trace}.
By Grauert-Riemenschneider we have $Rf_*\omega_X = f_*\omega_X$
thus the trace map indeed produces a map $\Gamma(X, \omega_X) \to \omega_A$.
By duality we have $Rf_*\omega_X = R\Hom_A(Rf_*\mathcal{O}_X, \omega_A)$
(this uses that $\omega_X[2]$ is the dualizing complex on $X$
normalized relative to $\omega_A[2]$,
see Dualizing Complexes, Lemma \ref{dualizing-lemma-duality-bootstrap}
or more directly Section \ref{dualizing-section-duality} or even more directly
Lemma \ref{dualizing-lemma-proper-noetherian-relative}).
The distinguished triangle
$$
A \to Rf_*\mathcal{O}_X \to R^1f_*\mathcal{O}_X[-1] \to A[1]
$$
is transformed by $R\Hom_A(-, \omega_A)$ into the short exact sequence
$$
0 \to f_*\omega_X \to \omega_A \to
\text{Ext}_A^2(R^1f_*\mathcal{O}_X, \omega_A) \to 0
$$
(and $\text{Ext}_A^i(R^1f_*\mathcal{O}_X, \omega_A) = 0$ for $i \not = 2$;
this will follow from the discussion below as well).
Since $R^1f_*\mathcal{O}_X$ is supported in $\{\mathfrak m\}$, the
local duality theorem tells us that
$$
\text{Ext}_A^2(R^1f_*\mathcal{O}_X, \omega_A) =
\text{Ext}_A^0(R^1f_*\mathcal{O}_X, \omega_A[2]) =
\Hom_A(R^1f_*\mathcal{O}_X, E)
$$
is the Matlis dual of $R^1f_*\mathcal{O}_X$ (and the other
ext groups are zero), see
Dualizing Complexes, Lemma \ref{dualizing-lemma-special-case-local-duality}.
By the equivalence of categories inherent in Matlis duality
(Dualizing Complexes, Proposition \ref{dualizing-proposition-matlis}),
if $R^1f_*\mathcal{O}_X$ is not annihilated by $d$,
then neither is the $\text{Ext}^2$ above. Hence we see that
$H^1(X, \mathcal{O}_X)$ is annihilated by $d$. Thus the required
boundedness follows from Lemma \ref{lemma-bound-a-torsion} and
(\ref{equation-a-torsion}).
\end{proof}
\begin{lemma}
\label{lemma-compare-differentials-dualizing}
Let $p$ be a prime number.
Let $A$ be a regular local ring of dimension $2$ and characteristic $p$.
Let $A_0 \subset A$ be a subring such that $\Omega_{A/A_0}$ is free
of rank $r < \infty$. Set $\omega_A = \Omega^r_{A/A_0}$. If $X \to \Spec(A)$
is the result of a sequence of blowups in closed points, then
there exists a map
$$
\varphi_X : (\Omega^r_{X/\Spec(A_0)})^{**} \longrightarrow \omega_X
$$
extending the given identification in the generic point.
\end{lemma}
\begin{proof}
Observe that $A$ is Gorenstein (Dualizing Complexes,
Lemma \ref{dualizing-lemma-regular-gorenstein})
and hence the invertible module $\omega_A$ does indeed serve
as a dualizing module. Moreover, any $X$ as in the lemma
has an invertible dualizing module $\omega_X$ as $X$ is regular
(hence Gorenstein) and proper over $A$, see
Remark \ref{remark-dualizing-setup} and
Lemma \ref{lemma-blowup-regular}.
Suppose we have constructed the map
$\varphi_X : (\Omega^r_{X/A_0})^{**} \to \omega_X$
and suppose that $b : X' \to X$ is a blow up in a closed point.
Set $\Omega^r_X = (\Omega^r_{X/A_0})^{**}$ and
$\Omega^r_{X'} = (\Omega^r_{X'/A_0})^{**}$. Since $\omega_{X'} = b^!(\omega_X)$
a map $\Omega^r_{X'} \to \omega_{X'}$ is the same thing as a map
$Rb_*(\Omega^r_{X'}) \to \omega_X$. See discussion in
Remark \ref{remark-dualizing-setup} and
Dualizing Complexes, Section \ref{dualizing-section-duality}.
Thus in turn it suffices to produce a map
$$
Rb_*(\Omega^r_{X'}) \longrightarrow \Omega^r_X
$$
The sheaves $\Omega^r_{X'}$ and $\Omega^r_X$ are invertible, see
Divisors, Lemma \ref{divisors-lemma-reflexive-over-regular-dim-2}.
Consider the exact sequence
$$
b^*\Omega_{X/A_0} \to \Omega_{X'/A_0} \to \Omega_{X'/X} \to 0
$$
A local calculation shows that $\Omega_{X'/X}$ is isomorphic
to an invertible module on the exceptional divisor $E$, see
Lemma \ref{lemma-differentials-of-blowup}. It follows that
either
$$
\Omega^r_{X'} \cong (b^*\Omega^r_X)(E)
\quad\text{or}\quad
\Omega^r_{X'} \cong b^*\Omega^r_X
$$
see Divisors, Lemma \ref{divisors-lemma-wedge-product-ses}.
(The second possibility never happens in characteristic zero, but
can happen in characteristic $p$.) In both cases we see that
$R^1b_*(\Omega^r_{X'}) = 0$ and $b_*(\Omega^r_{X'}) = \Omega^r_X$ by
Lemma \ref{lemma-cohomology-of-blowup}.
\end{proof}
\begin{lemma}
\label{lemma-go-up-degree-p}
Let $p$ be a prime number. Let $A$ be a complete regular local ring of
dimension $2$ and characteristic $p$. Let $K = f.f.(A) \subset L$ be a
degree $p$ inseparable extension and let $B \subset L$ be the integral
closure of $A$. Then reduction to rational singularities is possible for $B$.
\end{lemma}
\begin{proof}
We have $A = k[[x, y]]$. Write $L = K[x]/(x^p - f)$ for some $f \in A$
and denote $g \in B$ the congruence class of $x$, i.e., the element such
that $g^p = f$. By
More on Algebra, Lemma \ref{more-algebra-lemma-power-series-ring-subfields}
there exists a subfield $k^p \subset k' \subset k$ with
$p^e = [k : k'] < \infty$
such that $f$ is not contained in the fraction field $K_0$ of
$A_0 = k'[[x^p, y^p]] \subset A$. Then
$$
\Omega_{A/A_0} =
A \otimes_k \Omega_{k/k'} \oplus A \text{d}x \oplus A \text{d}y
$$
is finite free of rank $e + 2$. Set $\omega_A = \Omega^{e + 2}_{A/A_0}$.
Consider the canonical map
$$
\text{Tr} :
\Omega^{e + 2}_{B/A_0}
\longrightarrow
\Omega^{e + 2}_{A/A_0} = \omega_A
$$
of Lemma \ref{lemma-trace-extends}. By duality this determines a map
$$
c : \Omega^{e + 2}_{B/A_0} \to \omega_B = \Hom_A(B, \omega_A)
$$
Claim: the cokernel of $c$ is annihilated by a nonzero element of $B$.
\medskip\noindent
Since $\text{d}f$ is nonzero in $\Omega_{A/A_0}$
(Algebra, Lemma \ref{algebra-lemma-derivative-zero-pth-power}) we can find
$\eta_1, \ldots, \eta_{e + 1} \in \Omega_{A/A_0}$ such that
$\theta = \eta_1 \wedge \ldots \wedge \eta_{e + 1} \wedge \text{d}f$ is
nonzero in $\omega_A = \Omega^{e + 2}_{A/A_0}$. To prove the claim we
will construct elements $\omega_i$ of $\Omega^{e + 2}_{B/A_0}$,
$i = 0, \ldots, p - 1$ which are mapped to
$\varphi_i \in \omega_B = \Hom_A(B, \omega_A)$
with $\varphi_i(g^j) = \delta_{ij}\theta$ for $j = 0, \ldots, p - 1$.
Since $\{1, g, \ldots, g^{p - 1}\}$ is a basis for $L/K$ this
proves the claim. We set
$\eta = \eta_1 \wedge \ldots \wedge \eta_{e + 1}$
so that $\theta = \eta \wedge \text{d}f$.
Set $\omega_i = \eta \wedge g^{p - 1 - i}\text{d}g$. Then
by construction we have
$$
\varphi_i(g^j) = \text{Tr}(g^j \eta \wedge g^{p - 1 - i}\text{d}g) =
\text{Tr}(\eta \wedge g^{p - 1 - i + j}\text{d}g) = \delta_{ij} \theta
$$
by the explicit description of the trace map in Lemma \ref{lemma-trace-higher}.
\medskip\noindent
Let $Y \to \Spec(B)$ be a normal modification. Exactly as in the proof of
Lemma \ref{lemma-go-up-separable} we can reduce to the case where $Y$
is finite over a modification $X$ of $\Spec(A)$. Arguing as in the proof
of Lemma \ref{lemma-regular-rational} we may even assume that $X = X_n$ where
$$
X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X
$$
is a sequence of blowing ups in closed points. By
Lemma \ref{lemma-trace-extends} we obtain the first arrow in
$$
\pi_*(\Omega^{e + 2}_{Y/A_0})
\xrightarrow{\text{Tr}}
(\Omega^{e + 2}_{X/A_0})^{**}
\xrightarrow{\varphi_X}
\omega_X
$$
and the second arrow is from
Lemma \ref{lemma-compare-differentials-dualizing}.
By duality this corresponds to a map
$$
c_Y : \Omega^{e + 2}_{Y/A_0} \longrightarrow \omega_Y
$$
extending the map $c$ above. Hence we see that the image of
$\Gamma(Y, \omega_Y) \to \omega_B$ contains the image of $c$.
By our claim we see that the cokernel is annihilated by
a fixed nonzero element of $B$. We conclude by
Lemma \ref{lemma-bound-dualizing-implies-bound}.
\end{proof}
\section{Rational singularities}
\label{section-rational-singularities}
\noindent
In this section we reduce from rational singular points to
Gorenstein rational singular points. See \cite{Lipman-rational} and
\cite{Mattuck}.
\begin{situation}
\label{situation-rational}
Here $(A, \mathfrak m, \kappa)$ be a local normal Nagata domain of
dimension $2$ which defines a rational singularity. Let $s$ be the closed
point of $S = \Spec(A)$ and $U = S \setminus \{s\}$. Let $f : X \to S$
be a modification with $X$ normal.
We denote $C_1, \ldots, C_r$ the irreducible
components of the special fibre $X_s$ of $f$.
\end{situation}
\begin{lemma}
\label{lemma-globally-generated}
In Situation \ref{situation-rational}.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then
\begin{enumerate}
\item $H^p(X, \mathcal{F}) = 0$ for $p \not \in \{0, 1\}$, and
\item $H^1(X, \mathcal{F}) = 0$ if $\mathcal{F}$ is globally generated.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (1) follows from Cohomology of Schemes, Lemma
\ref{coherent-lemma-higher-direct-images-zero-above-dimension-fibre}.
If $\mathcal{F}$ is globally generated, then there is a surjection
$\bigoplus_{i \in I} \mathcal{O}_X \to \mathcal{F}$. By part (1)
and the long exact sequence of cohomology this
induces a surjection on $H^1$. Since $H^1(X, \mathcal{O}_X) = 0$
as $S$ has a rational singularity, and since $H^1(X, -)$ commutes
with direct sums
(Cohomology, Lemma \ref{cohomology-lemma-quasi-separated-cohomology-colimit})
we conclude.
\end{proof}
\begin{lemma}
\label{lemma-sections-powers-I-rational}
In Situation \ref{situation-rational} assume
$E = X_s$ is an effective Cartier divisor.
Let $\mathcal{I}$ be the ideal sheaf of $E$. Then
$H^0(X, \mathcal{I}^n) = \mathfrak m^n$ and
$H^1(X, \mathcal{I}^n) = 0$.
\end{lemma}
\begin{proof}
We have $H^0(X, \mathcal{O}_X) = A$, see discussion following
Situation \ref{situation-vanishing}. Then
$\mathfrak m \subset H^0(X, \mathcal{I}) \subset H^0(X, \mathcal{O}_X)$.
The second inclusion is not an equality as $X_s \not = \emptyset$.
Thus $H^0(X, \mathcal{I}) = \mathfrak m$.
As $\mathcal{I}^n = \mathfrak m^n\mathcal{O}_X$ our
Lemma \ref{lemma-globally-generated} shows that $H^1(X, \mathcal{I}^n) = 0$.
\medskip\noindent
Choose generators $x_1, \ldots, x_{\mu + 1}$ of $\mathfrak m$. These define
global sections of $\mathcal{I}$ which generate it. Hence
a short exact sequence
$$
0 \to \mathcal{F} \to \mathcal{O}_X^{\oplus \mu + 1} \to \mathcal{I} \to 0
$$
Then $\mathcal{F}$ is a finite locally free $\mathcal{O}_X$-module
of rank $\mu$ and $\mathcal{F} \otimes \mathcal{I}$ is globally
generated by Constructions, Lemma
\ref{constructions-lemma-globally-generated-omega-twist-1}.
Hence $\mathcal{F} \otimes \mathcal{I}^n$
is globally generated for all $n \geq 1$. Thus for $n \geq 2$ we can
consider the exact sequence
$$
0 \to \mathcal{F} \otimes \mathcal{I}^{n - 1} \to
(\mathcal{I}^{n - 1})^{\oplus \mu + 1} \to
\mathcal{I}^n \to 0
$$
Applying the long exact sequence of cohomology using that
$H^1(X, \mathcal{F} \otimes \mathcal{I}^{n - 1}) = 0$ by
Lemma \ref{lemma-globally-generated}
we obtain that every
element of $H^0(X, \mathcal{I}^n)$ is of the form $\sum x_i a_i$
for some $a_i \in H^0(X, \mathcal{I}^{n - 1})$. This shows that
$H^0(X, \mathcal{I}^n) = \mathfrak m^n$ by induction.
\end{proof}
\begin{lemma}
\label{lemma-blow-up-normal-rational}
In Situation \ref{situation-rational}
the blow up of $\Spec(A)$ in $\mathfrak m$ is normal.
\end{lemma}
\begin{proof}
Let $X' \to \Spec(A)$ be the blow up, in other words
$$
X' = \text{Proj}(A \oplus \mathfrak m \oplus \mathfrak m^2 \oplus \ldots).
$$
is the Proj of the Rees algebra. This in particular shows that
$X'$ is integral and that $X' \to \Spec(A)$ is a projective
modification. Let $X$ be the normalization of $X'$.
Since $A$ is Nagata, we see that $\nu : X \to X'$ is finite
(Morphisms, Lemma \ref{morphisms-lemma-nagata-normalization}).
Let $E' \subset X'$ be the exceptional divisor and let $E \subset X$
be the inverse image. Let $\mathcal{I}' \subset \mathcal{O}_{X'}$
and $\mathcal{I} \subset \mathcal{O}_X$ be their ideal sheaves.
Recall that $\mathcal{I}' = \mathcal{O}_{X'}(1)$
(Divisors, Lemma \ref{divisors-lemma-blowing-up-projective}).
Observe that $\mathcal{I} = \nu^*\mathcal{I}'$ and that $E$ is an
effective Cartier divisor (Divisors, Lemma
\ref{divisors-lemma-pullback-effective-Cartier-defined}).
We are trying to show that $\nu$ is an isomorphism. As $\nu$ is finite,
it suffices to show that $\mathcal{O}_{X'} \to \nu_*\mathcal{O}_X$
is an isomorphism. If not, then we can find an $n \geq 0$ such that
$$
H^0(X', (\mathcal{I}')^n) \not =
H^0(X', (\nu_*\mathcal{O}_X) \otimes (\mathcal{I}')^n)
$$
for example because we can recover quasi-coherent $\mathcal{O}_{X'}$-modules
from their associated graded modules, see
Properties, Lemma \ref{properties-lemma-ample-quasi-coherent}.
By the projection formula we have
$$
H^0(X', (\nu_*\mathcal{O}_X) \otimes (\mathcal{I}')^n) =
H^0(X, \nu^*(\mathcal{I}')^n) =
H^0(X, \mathcal{I}^n) = \mathfrak m^n
$$
the last equality by Lemma \ref{lemma-sections-powers-I-rational}.
On the other hand, there is clearly an injection
$\mathfrak m^n \to H^0(X', (\mathcal{I}')^n)$. Since
$H^0(X', (\mathcal{I}')^n)$ is torsion free we conclude equality holds
for all $n$, hence $X = X'$.
\end{proof}
\begin{lemma}
\label{lemma-cohomology-blow-up-rational}
In Situation \ref{situation-rational}.
Let $X$ be the blow up of $\Spec(A)$ in $\mathfrak m$. Let $E \subset X$
be the exceptional divisor. With $\mathcal{O}_X(1) = \mathcal{I}$ as
usual and $\mathcal{O}_E(1) = \mathcal{O}_X(1)|_E$ we have
\begin{enumerate}
\item $E$ is a proper Cohen-Macaulay curve over $\kappa$.
\item $\mathcal{O}_E(1)$ is very ample
\item $\deg(\mathcal{O}_E(1)) \geq 1$ and equality holds only if
$A$ is a regular local ring,
\item $H^1(E, \mathcal{O}_E(n)) = 0$ for $n \geq 0$, and
\item $H^0(E, \mathcal{O}_E(n)) = \mathfrak m^n/\mathfrak m^{n + 1}$
for $n \geq 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
Since $\mathcal{O}_X(1)$ is very ample by construction, we see that
its restriction to the special fibre $E$ is very ample as well.
By Lemma \ref{lemma-blow-up-normal-rational} the scheme $X$ is normal.
Then $E$ is Cohen-Macaulay by
Divisors, Lemma \ref{divisors-lemma-normal-effective-Cartier-divisor-S1}.
Lemma \ref{lemma-sections-powers-I-rational} applies and we obtain
(4) and (5) from the exact sequences
$$
0 \to \mathcal{I}^{n + 1} \to \mathcal{I}^n \to i_*\mathcal{O}_E(n) \to 0
$$
and the long exact cohomology sequence. In particular, we see that
$$
\deg(\mathcal{O}_E(1)) = \chi(E, \mathcal{O}_E(1)) - \chi(E, \mathcal{O}_E) =
\dim(\mathfrak m/\mathfrak m^2) - 1
$$
by Varieties, Definition \ref{varieties-definition-degree-invertible-sheaf}.
Thus (3) follows as well.
\end{proof}
\begin{lemma}
\label{lemma-dualizing-rational}
In Situation \ref{situation-rational} assume $A$ has a
dualizing complex $\omega_A^\bullet$. With $\omega_X$ the dualizing
module of $X$, the trace map $H^0(X, \omega_X) \to \omega_A$ is an
isomorphism and consequently there is a canonical map
$f^*\omega_A \to \omega_X$.
\end{lemma}
\begin{proof}
By Grauert-Riemenschneider
(Proposition \ref{proposition-Grauert-Riemenschneider}) we see that
$Rf_*\omega_X = f_*\omega_X$. By duality we have a short exact
sequence
$$
0 \to f_*\omega_X \to \omega_A \to
\text{Ext}^2_A(R^1f_*\mathcal{O}_X, \omega_A) \to 0
$$
(for example see proof of Lemma \ref{lemma-bound-dualizing-implies-bound})
and since $A$ defines a rational singularity we obtain
$f_*\omega_X = \omega_A$.
\end{proof}
\begin{lemma}
\label{lemma-dualizing-blow-up-rational}
In Situation \ref{situation-rational} assume $A$ has a
dualizing complex $\omega_A^\bullet$ and is not regular.
Let $X$ be the blow up of $\Spec(A)$ in $\mathfrak m$ with
exceptional divisor $E \subset X$. Let $\omega_X$
be the dualizing module of $X$. Then
\begin{enumerate}
\item $\omega_E = \omega_X|_E \otimes \mathcal{O}_E(-1)$,
\item $H^1(X, \omega_X(n)) = 0$ for $n \geq 0$,
\item the map $f^*\omega_A \to \omega_X$ of
Lemma \ref{lemma-dualizing-rational} is surjective.
\end{enumerate}
\end{lemma}
\begin{proof}
We will use the results of Lemma \ref{lemma-cohomology-blow-up-rational}
without further mention. Observe that
$\omega_E = \omega_X|_E \otimes \mathcal{O}_E(-1)$
by Dualizing Complexes, Lemmas
\ref{dualizing-lemma-sheaf-with-exact-support-effective-Cartier} and
\ref{dualizing-lemma-twisted-inverse-image-closed}. Thus
$\omega_X|_E = \omega_E(1)$. Consider the short exact sequences
$$
0 \to \omega_X(n + 1) \to \omega_X(n) \to i_*\omega_E(n + 1) \to 0
$$
By Algebraic Curves, Lemma \ref{curves-lemma-vanishing-twist}
we see that $H^1(E, \omega_E(n + 1)) = 0$ for $n \geq 0$.
Thus we see that the maps
$$
\ldots \to H^1(X, \omega_X(2)) \to H^1(X, \omega_X(1)) \to H^1(X, \omega_X)
$$
are surjective. Since $H^1(X, \omega_X(n))$ is zero for $n \gg 0$
(Cohomology of Schemes, Lemma \ref{coherent-lemma-kill-by-twisting})
we conclude that (2) holds.
\medskip\noindent
By Algebraic Curves, Lemma
\ref{curves-lemma-tensor-omega-with-globally-generated-invertible}
we see that $\omega_X|_E = \omega_E \otimes \mathcal{O}_E(1)$
is globally generated. Since we seen above that
$H^1(X, \omega_X(1)) = 0$ the map $H^0(X, \omega_X) \to H^0(E, \omega_X|_E)$
is surjective. We conclude that $\omega_X$ is globally generated
hence (3) holds because $\Gamma(X, \omega_X) = \omega_A$ is used
in Lemma \ref{lemma-dualizing-rational} to define the map.
\end{proof}
\begin{lemma}
\label{lemma-rational-to-gorenstein}
Let $(A, \mathfrak m, \kappa)$ be a local normal Nagata domain of
dimension $2$ which defines a rational singularity. Assume $A$ has
a dualizing complex. Then there exists a finite sequence of blowups in
singular closed points
$$
X = X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \Spec(A)
$$
such that $X_i$ is normal for each $i$ and such that
the dualizing sheaf $\omega_X$ of $X$ is an invertible
$\mathcal{O}_X$-module.
\end{lemma}
\begin{proof}
The dualizing module $\omega_A$ is a finite $A$-module whose stalk at
the generic point is invertible. Namely, $\omega_A \otimes_A K$
is a dualizing module for the fraction field $K$ of $A$, hence has
rank $1$. Thus there exists a blowup $b : Y \to \Spec(A)$ such that
the strict transform of $\omega_A$ with respect to $b$ is an invertible
$\mathcal{O}_Y$-module. This follows from the definition of strict
transform in
Divisors, Definition \ref{divisors-definition-strict-transform},
the description of the strict transform of quasi-coherent modules in
Properties, Lemma
\ref{properties-lemma-sections-supported-on-closed-subset}, and
More on Algebra, Lemma \ref{more-algebra-lemma-blowup-module}.
By Lemma \ref{lemma-dominate-by-normalized-blowing-up}
we can choose a sequence of normalized blowups
$$
X_n \to X_{n - 1} \to \ldots \to X_1 \to \Spec(A)
$$
such that $X_n$ dominates $Y$. By Lemma \ref{lemma-blow-up-normal-rational}
and arguing by induction each $X_i \to X_{i - 1}$ is simply a blowing up.
\medskip\noindent
We claim that $\omega_{X_n}$ is invertible. Since $\omega_{X_n}$
is a coherent $\mathcal{O}_{X_n}$-module, it suffices to see its stalks
are invertible modules. If $x \in X_n$ is a regular point, then this is
clear from the fact that regular schemes are
Gorenstein (Dualizing Complexes, Lemma
\ref{dualizing-lemma-regular-gorenstein}). If $x$ is a singular point of
$X_n$, then each of the images $x_i \in X_i$ of $x$ is a singular point
(because the blowup of a regular point is regular by
Lemma \ref{lemma-blowup-regular}).
Consider the canonical map $f_n^*\omega_A \to \omega_{X_n}$ of
Lemma \ref{lemma-dualizing-rational}. For each $i$ the morphism
$X_{i + 1} \to X_i$ is either a blowup of $x_i$ or an isomorphism
at $x_i$. Since $x_i$ is always a singular point, it follows from
Lemma \ref{lemma-dualizing-blow-up-rational}
and induction that the maps $f_i^*\omega_A \to \omega_{X_i}$
is always surjective on stalks at $x_i$. Hence
$$
(f_n^*\omega_A)_x \longrightarrow \omega_{X_n, x}
$$
is surjective. On the other hand, by our choice of $b$ the quotient
of $f_n^*\omega_A$ by its torsion submodule is an invertible module
$\mathcal{L}$. Moreover, the dualizing module is torsion free
(Dualizing Complexes, Lemma \ref{dualizing-lemma-dualizing-module}).
It follows that $\mathcal{L}_x \cong \omega_{X_n, x}$ and the proof is
complete.
\end{proof}
\section{Formal arcs}
\label{section-arcs}
\noindent
Let $X$ be a locally Noetherian scheme. In this section we say that a
{\it formal arc} in $X$ is a morphism $a : T \to X$ where $T$ is the
spectrum of a complete discrete valuation ring $R$ whose residue field
$\kappa$ is identified with the residue field of the image $p$ of the
closed point of $\Spec(R)$. Let us say that the formal arc $a$ is
{\it centered at $p$} in this case. We say the formal arc $T \to X$
is {\it nonsingular} if the induced map
$\mathfrak m_p/\mathfrak m_p^2 \to \mathfrak m_R/\mathfrak m_R^2$
is surjective.
\medskip\noindent
Let $a : T \to X$, $T = \Spec(R)$ be a nonsingular formal arc centered
at a closed point $p$ of $X$. Assume $X$ is locally Noetherian.
Let $b : X_1 \to X$ be the blowing up of $X$ at $x$.
Since $a$ is nonsingular, we see that there is an element
$f \in \mathfrak m_p$ which maps to a uniformizer in $R$.
In particular, we find that the generic point of $T$ maps to
a point of $X$ not equal to $p$. In other words, with $K = f.f.(R)$
the fraction field, the restriction of $a$ defines a morphism
$\Spec(K) \to X \setminus \{p\}$. Since the morphism $b$ is
proper and an isomorphism over $X \setminus \{x\}$ we can apply
the valuative criterion of properness to obtain a unique morphism
$a_1$ making the following diagram commute
$$
\xymatrix{
T \ar[r]_{a_1} \ar[rd]_a & X_1 \ar[d]^{b} \\
& X
}
$$
Let $p_1 \in X_1$ be the image of the closed point of $T$. Observe that
$p_1$ is a closed point as it is a $\kappa = \kappa(p)$-rational point
on the fibre of $X_1 \to X$ over $x$. Since we have a factorization
$$
\mathcal{O}_{X, x} \to \mathcal{O}_{X_1, p_1} \to R
$$
we see that $a_1$ is a nonsingular formal arc as well.
\medskip\noindent
We can repeat the process and obtain a sequence of blowing ups
$$
\xymatrix{
T \ar[d]_a \ar[rd]_{a_1} \ar[rrd]_{a_2} \ar[rrrd]^{a_3} \\
(X, p) & (X_1, p_1) \ar[l] & (X_2, p_2) \ar[l] &
(X_3, p_3) \ar[l] & \ar[l] \ldots
}
$$
This kind of sequence of blowups can be characterized as follows.
\begin{lemma}
\label{lemma-sequence-blowups}
Let $X$ be a locally Noetherian scheme. Let
$$
(X, p) = (X_0, p_0) \leftarrow (X_1, p_1) \leftarrow (X_2, p_2) \leftarrow
(X_3, p_3) \leftarrow \ldots
$$
be a sequence of blowups such that
\begin{enumerate}
\item $p_i$ is closed, maps to $p_{i - 1}$, and
$\kappa(p_i) = \kappa(p_{i - 1})$,
\item there exists an $x_1 \in \mathfrak m_p$ whose image
in $\mathfrak m_{p_i}$, $i > 0$ defines the exceptional divisor
$E_i \subset X_i$.
\end{enumerate}
Then the sequence is obtained from a nonsingular arc $a : T \to X$
as above.
\end{lemma}
\begin{proof}
Let us write $\mathcal{O}_n = \mathcal{O}_{X_n, p_n}$
and $\mathcal{O} = \mathcal{O}_{X, p}$. Denote
$\mathfrak m \subset \mathcal{O}$ and $\mathfrak m_n \subset \mathcal{O}_n$
the maximal ideals.
\medskip\noindent
We claim that $x_1^t \not \in \mathfrak m_n^{t + 1}$.
Namely, if this were the case, then in the local ring
$\mathcal{O}_{n + 1}$ the element $x_1^t$ would be in the ideal of
$(t + 1)E_{n + 1}$.
This contradicts the assumption that $x_1$ defines $E_{n + 1}$.
\medskip\noindent
For every $n$ choose generators $y_{n, 1}, \ldots, y_{n, t_n}$
for $\mathfrak m_n$. As
$\mathfrak m_n \mathcal{O}_{n + 1} = x_1\mathcal{O}_{n + 1}$
by assumption (2), we can write $y_{n, i} = a_{n, i} x_1$
for some $a_{n, i} \in \mathcal{O}_{n + 1}$. Since
the map $\mathcal{O}_n \to \mathcal{O}_{n + 1}$ defines
an isomorphism on residue fields by (1) we can choose
$c_{n, i} \in \mathcal{O}_n$ having the same residue class as
$a_{n, i}$. Then we see that
$$
\mathfrak m_n = (x_1, z_{n, 1}, \ldots, z_{n, t_n}),
\quad z_{n, i} = y_{n, i} - c_{n, i} x_1
$$
and the elements $z_{n, i}$ map to elements of $\mathfrak m_{n + 1}^2$
in $\mathcal{O}_{n + 1}$.
\medskip\noindent
Let us consider
$$
J_n = \Ker(\mathcal{O} \to \mathcal{O}_n/\mathfrak m_n^{n + 1})
$$
We claim that $\mathcal{O}/J_n$ has length $n + 1$ and that
$\mathcal{O}/(x_1) + J_n$ equals the residue field. For $n = 0$
this is immediate. Assume the statement holds for $n$.
Let $f \in J_n$. Then in $\mathcal{O}_n$ we have
$$
f = a x_1^{n + 1} + x_1^n A_1(z_{n, i}) +
x_1^{n - 1} A_2(z_{n, i}) + \ldots + A_{n + 1}(z_{n, i})
$$
for some $a \in \mathcal{O}_n$ and some $A_i$ homogeneous of degree $i$
with coefficients in $\mathcal{O}_n$. Since $\mathcal{O} \to \mathcal{O}_n$
identifies residue fields, we may choose $a \in \mathcal{O}$
(argue as in the construction of $z_{n, i}$ above).
Taking the image in
$\mathcal{O}_{n + 1}$ we see that $f$ and $a x_1^{n + 1}$
have the same image modulo $\mathfrak m_{n + 1}^{n + 2}$.
Since $x_n^{n + 1} \not \in \mathfrak m_{n + 1}^{n + 2}$
it follows that $J_n/J_{n + 1}$ has length $1$ and the claim is true.
\medskip\noindent
Consider $R = \lim \mathcal{O}/J_n$. This is a quotient of
the $\mathfrak m$-adic completion of $\mathcal{O}$ hence it is
a complete Noetherian local ring. On the other hand, it is
not finite length and $x_1$ generates the maximal ideal.
Thus $R$ is a complete discrete valuation ring.
The map $\mathcal{O} \to R$ lifts to a local homomorphism
$\mathcal{O}_n \to R$ for every $n$. There are two ways to show this:
(1) for every $n$ one can use a similar procedure
to construct $\mathcal{O}_n \to R_n$ and then one can
show that $\mathcal{O} \to \mathcal{O}_n \to R_n$ factors
through an isomorphism $R \to R_n$, or (2) one can use
Divisors, Lemma \ref{divisors-lemma-characterize-affine-blowup}
to show that $\mathcal{O}_n$ is a localization of a repeated
affine blowup algebra to explicitly construct a map $\mathcal{O}_n \to R$.
Having said this it is clear that our sequence of blow ups
comes from the nonsingular arc $a : T = \Spec(R) \to X$.
\end{proof}
\noindent
The following lemma is a kind of N\'eron desingularization lemma.
\begin{lemma}
\label{lemma-sequence-blowups-along-arc-becomes-nonsingular}
Let $(A, \mathfrak m, \kappa)$ be a Noetherian local domain of
dimension $2$. Let $A \to R$ be a surjection onto a
complete discrete valuation ring.
This defines a nonsingular arc $a : T = \Spec(R) \to \Spec(A)$. Let
$$
\Spec(A) = X_0 \leftarrow X_1 \leftarrow X_2 \leftarrow X_3 \leftarrow \ldots
$$
be the sequence of blowing ups constructed from $a$.
If $A_\mathfrak p$ is a regular local ring where
$\mathfrak p = \Ker(A \to R)$, then
for some $i$ the scheme $X_i$ is regular at $x_i$.
\end{lemma}
\begin{proof}
Let $x_1 \in \mathfrak m$ map to a uniformizer of $R$.
Observe that $\kappa(\mathfrak p) = K = f.f.(R)$ is the
fraction field of $R$. Write $\mathfrak p = (x_2, \ldots, x_r)$
with $r$ minimal. If $r = 2$, then $\mathfrak m = (x_1, x_2)$
and $A$ is regular and the lemma is true. Assume $r > 2$.
After renumbering if necessary,
we may assume that $x_2$ maps to a uniformizer of $A_\mathfrak p$.
Then $\mathfrak p/\mathfrak p^2 + (x_2)$ is annihilated by a power
of $x_1$. For $i > 2$ we can find $n_i \geq 0$ and $a_i \in A$
such that
$$
x_1^{n_i} x_i - a_i x_2 = \sum\nolimits_{2 \leq j \leq k} a_{jk} x_jx_k
$$
for some $a_{jk} \in A$. If $n_i = 0$ for some $i$, then we can remove
$x_i$ from the list of generators of $\mathfrak p$ and we win by
induction on $r$. If for some $i$ the element $a_i$ is a unit, then
we can remove $x_2$ from the list of generators of $\mathfrak p$
and we win in the same manner. Thus either
$a_i \in \mathfrak p$ or $a_i = u_i x_1^{m_1} \bmod \mathfrak p$
for some $m_1 > 0$ and unit $u_i \in A$. Thus we have either
$$
x_1^{n_i} x_i = \sum\nolimits_{2 \leq j \leq k} a_{jk} x_jx_k
\quad\text{or}\quad
x_1^{n_i} x_i - u_i x_1^{m_i} x_2 =
\sum\nolimits_{2 \leq j \leq k} a_{jk} x_jx_k
$$
We will prove that after blowing up the integers $n_i$, $m_i$
decrease which will finish the proof.
\medskip\noindent
Let us see what happens with these equations on the affine blowup
algebra $A' = A[\mathfrak m/x_1]$. As $\mathfrak m = (x_1, \ldots, x_r)$
we see that $A'$ is generated over $R$ by $y_i = x_i/x_1$ for $i \geq 2$.
Clearly $A \to R$ extends to $A' \to R$ with kernel
$(y_2, \ldots, y_r)$. Then we see that either
$$
x_1^{n_i - 1} y_i = \sum\nolimits_{2 \leq j \leq k} a_{jk} y_jy_k
\quad\text{or}\quad
x_1^{n_i - 1} y_i - u_i x_1^{m_1 - 1} y_2 =
\sum\nolimits_{2 \leq j \leq k} a_{jk} y_jy_k
$$
and the proof is complete.
\end{proof}
\section{Base change to the completion}
\label{section-aux}
\noindent
The following simple lemma will turn out to be a useful tool in what follows.
\begin{lemma}
\label{lemma-iso-completions}
Let $(A, \mathfrak m, \kappa)$ be a local ring with finitely generated
maximal ideal $\mathfrak m$. Let $X$ be a scheme over $A$.
Let $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$ where
$A^\wedge$ is the $\mathfrak m$-adic completion of $A$.
For a point $q \in Y$ with image $p \in X$ lying
over the closed point of $\Spec(A)$ the
local ring map $\mathcal{O}_{X, p} \to \mathcal{O}_{Y, q}$
induces an isomorphism on completions.
\end{lemma}
\begin{proof}
We may assume $X$ is affine. Then we may write $X = \Spec(B)$.
Let $\mathfrak q \subset B' = B \otimes_A A^\wedge$ be the
prime corresponding to $q$ and let $\mathfrak p \subset B$
be the prime ideal corresponding to $p$.
By Algebra, Lemma \ref{algebra-lemma-hathat-finitely-generated}
we have
$$
B'/(\mathfrak m^\wedge)^n B' =
A^\wedge/(\mathfrak m^\wedge)^n \otimes_A B =
A/\mathfrak m^n \otimes_A B = B/\mathfrak m^n B
$$
for all $n$. Since $\mathfrak m B \subset \mathfrak p$ and
$\mathfrak m^\wedge B' \subset \mathfrak q$ we see that
$B/\mathfrak p^n$ and $B'/\mathfrak q^n$ are both
quotients of the ring displayed above by the $n$th power
of the same prime ideal. The lemma follows.
\end{proof}
\begin{lemma}
\label{lemma-port-regularity-to-completion}
Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring.
Let $X \to \Spec(A)$ be a morphism which is locally of finite type.
Set $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$. Let $y \in Y$ with
image $x \in X$. Then
\begin{enumerate}
\item if $\mathcal{O}_{Y, y}$ is regular, then $\mathcal{O}_{X, x}$
is regular,
\item if $y$ is in the closed fibre, then $\mathcal{O}_{Y, y}$ is regular
$\Leftrightarrow \mathcal{O}_{X, x}$ is regular, and
\item If $X$ is proper over $A$, then $X$ is regular
if and only if $Y$ is regular.
\end{enumerate}
\end{lemma}
\begin{proof}
Since $A \to A^\wedge$ is faithfully flat
(Algebra, Lemma \ref{algebra-lemma-completion-faithfully-flat}),
we see that $Y \to X$ is flat. Hence (1) by
Algebra, Lemma \ref{algebra-lemma-descent-regular}.
Lemma \ref{lemma-iso-completions} shows the morphism $Y \to X$
induces an isomorphism on complete local rings at points
of the special fibres. Thus (2) by
More on Algebra, Lemma \ref{more-algebra-lemma-completion-regular}.
If $X$ is proper over $A$, then $Y$ is proper over $A^\wedge$
(Morphisms, Lemma \ref{morphisms-lemma-base-change-proper})
and we see every closed point of $X$ and $Y$ lies in the closed fibre.
Thus we see that $Y$ is a regular scheme if and only if $X$ is so by
Properties, Lemma \ref{properties-lemma-characterize-regular}.
\end{proof}
\begin{lemma}
\label{lemma-descend-admissible-blowup}
Let $(A, \mathfrak m)$ be a Noetherian local ring with completion $A^\wedge$.
Let $U \subset \Spec(A)$ and $U^\wedge \subset \Spec(A^\wedge)$ be the
punctured spectra. If $Y \to \Spec(A^\wedge)$ is a $U^\wedge$-admissible
blowup, then there exists a $U$-admissible blowup $X \to \Spec(A)$
such that $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$.
\end{lemma}
\begin{proof}
By definition there exists an ideal $J \subset A^\wedge$ such that
$V(J) = \{\mathfrak m A^\wedge\}$ and such that $Y$ is the blowup
of $S^\wedge$ in the closed subscheme defined by $J$, see
Divisors, Definition \ref{divisors-definition-admissible-blowup}.
Since $A^\wedge$ is Noetherian this implies
$\mathfrak m^n A^\wedge \subset J$ for some $n$.
Since $A^\wedge/\mathfrak m^n A^\wedge = A/\mathfrak m^n$
we find an ideal $\mathfrak m^n \subset I \subset A$
such that $J = I A^\wedge$. Let $X \to S$ be the blowup in $I$.
Since $A \to A^\wedge$ is flat
we conclude that the base change of $X$ is $Y$ by
Divisors, Lemma \ref{divisors-lemma-flat-base-change-blowing-up}.
\end{proof}
\begin{lemma}
\label{lemma-blowup-still-good}
Let $(A, \mathfrak m, \kappa)$ be a Nagata local normal domain of
dimension $2$. Assume $A$ defines a rational singularity and that
the completion $A^\wedge$ of $A$ is normal. Then
\begin{enumerate}
\item $A^\wedge$ defines a rational singularity, and
\item if $X \to \Spec(A)$ is the blowing up in $\mathfrak m$, then
for a closed point $x \in X$ the completion $\mathcal{O}_{X, x}$ is normal.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $Y \to \Spec(A^\wedge)$ be a modification with $Y$ normal.
We have to show that $H^1(Y, \mathcal{O}_Y) = 0$. By Varieties, Lemma
\ref{varieties-lemma-modification-normal-iso-over-codimension-1}
$Y \to \Spec(A^\wedge)$ is an isomorphism over the punctured
spectrum $U^\wedge = \Spec(A^\wedge) \setminus \{\mathfrak m^\wedge\}$.
By Lemma \ref{lemma-dominate-by-scheme-modification}
there exists a $U^\wedge$-admissible blowup $Y' \to \Spec(A^\wedge)$
dominating $Y$. By Lemma \ref{lemma-descend-admissible-blowup}
we find there exists a $U$-admissible blowup $X \to \Spec(A)$
whose base change to $A^\wedge$ dominates $Y$.
Since $A$ is Nagata, we can replace $X$ by its normalization
after which $X \to \Spec(A)$ is a normal modification (but
possibly no longer a $U$-admissible blowup).
Then $H^1(X, \mathcal{O}_X) = 0$ as $A$ defines a rational
singularity. It follows that
$H^1(X \times_{\Spec(A)} \Spec(A^\wedge),
\mathcal{O}_{X \times_{\Spec(A)} \Spec(A^\wedge)}) = 0$
by flat base change (Cohomology of Schemes, Lemma
\ref{coherent-lemma-flat-base-change-cohomology}
and flatness of $A \to A^\wedge$ by
Algebra, Lemma \ref{algebra-lemma-completion-flat}).
We find that $H^1(Y, \mathcal{O}_Y) = 0$ by
Lemma \ref{lemma-exact-sequence}.
\medskip\noindent
Finally, let $X \to \Spec(A)$ be the blowing up of $\Spec(A)$
in $\mathfrak m$. Then $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$
is the blowing up of $\Spec(A^\wedge)$ in $\mathfrak m^\wedge$.
By Lemma \ref{lemma-blow-up-normal-rational} we see that both $Y$
and $X$ are normal. On the other hand, $A^\wedge$ is excellent
(More on Algebra, Proposition
\ref{more-algebra-proposition-ubiquity-excellent})
hence every affine open in $Y$ is the spectrum of an
excellent normal domain
(More on Algebra, Lemma \ref{more-algebra-lemma-finite-type-over-excellent}).
Thus for $y \in Y$ the ring map
$\mathcal{O}_{Y, y} \to \mathcal{O}_{Y, y}^\wedge$
is regular and by
More on Algebra, Lemma \ref{more-algebra-lemma-normal-goes-up}
we find that $\mathcal{O}_{Y, y}^\wedge$ is normal.
If $x \in X$ is a closed point of the special fibre,
then there is a unique closed point $y \in Y$ lying over $x$.
Since $\mathcal{O}_{X, x} \to \mathcal{O}_{Y, y}$ induces
an isomorphism on completions (Lemma \ref{lemma-iso-completions})
we conclude.
\end{proof}
\begin{lemma}
\label{lemma-formally-unramified}
Let $(A, \mathfrak m)$ be a local Noetherian ring. Let
$X$ be a scheme over $A$. Assume
\begin{enumerate}
\item $A$ is analytically unramified
(Algebra, Definition \ref{algebra-definition-analytically-unramified}),
\item $X$ is locally of finite type over $A$, and
\item $X \to \Spec(A)$ is \'etale at the generic points of irreducible
components of $X$.
\end{enumerate}
Then the normalization of $X$ is finite over $X$.
\end{lemma}
\begin{proof}
Since $A$ is analytically unramified it is reduced
by Algebra, Lemma \ref{algebra-lemma-analytically-unramified-easy}.
Since the normalization of $X$ depends only on the reduction
of $X$, we may replace $X$ by its reduction $X_{red}$; note
that $X_{red} \to X$ is an isomorphism over the open $U$ where
$X \to \Spec(A)$ is \'etale because $U$ is reduced
(Descent, Lemma \ref{descent-lemma-reduced-local-smooth})
hence condition (3) remains true after this replacement.
In addition we may and do assume that $X = \Spec(B)$ is affine.
\medskip\noindent
The map
$$
K = \prod\nolimits_{\mathfrak p \subset A\text{ minimal}} \kappa(\mathfrak p)
\longrightarrow
K^\wedge = \prod\nolimits_{\mathfrak p^\wedge \subset A^\wedge\text{ minimal}}
\kappa(\mathfrak p^\wedge)
$$
is injective because $A \to A^\wedge$ is faithfully flat
(Algebra, Lemma \ref{algebra-lemma-completion-faithfully-flat})
hence induces a surjective map between sets of minimal primes
(by going down for flat ring maps, see
Algebra, Section \ref{algebra-section-going-up}).
Both sides are finite products of fields as our rings are Noetherian.
Let $L = \prod_{\mathfrak q \subset B\text{ minimal}} \kappa(\mathfrak q)$.
Our assumption (3) implies that $L = B \otimes_A K$ and that
$K \to L$ is a finite \'etale ring map (this is true
because $A \to B$ is generically finite, for example use
Algebra, Lemma \ref{algebra-lemma-generically-finite}
or the more detailed results in Morphisms, Section
\ref{morphisms-section-generically-finite}).
Since $B$ is reduced we see that $B \subset L$.
This implies that
$$
C = B \otimes_A A^\wedge \subset
L \otimes_A A^\wedge = L \otimes_K K^\wedge = M
$$
Then $M$ is the total ring of fractions of $C$ and
is a finite product of fields as a finite separable
algebra over $K^\wedge$. It follows that $C$ is reduced
and that its normalization $C'$ is the integral closure of
$C$ in $M$. The normalization $B'$ of $B$ is the integral
closure of $B$ in $L$. By flatness of $A \to A^\wedge$
we obtain an injective map $B' \otimes_A A^\wedge \to M$ whose
image is contained in $C'$. Picture
$$
B' \otimes_A A^\wedge \longrightarrow C'
$$
As $A^\wedge$ is Nagata (by
Algebra, Lemma \ref{algebra-lemma-Noetherian-complete-local-Nagata}),
we see that $C'$ is finite over
$C = B \otimes_A A^\wedge$ (see
Algebra, Lemmas
\ref{algebra-lemma-Noetherian-complete-local-Nagata} and
\ref{algebra-lemma-nagata-in-reduced-finite-type-finite-integral-closure}).
As $C$ is Noetherian, we conclude that
$B' \otimes_A A^\wedge$ is finite over $C = B \otimes_A A^\wedge$.
Therefore by faithfully flat descent
(Algebra, Lemma \ref{algebra-lemma-descend-properties-modules})
we see that $B'$ is finite over $B$ which is what we had to show.
\end{proof}
\begin{lemma}
\label{lemma-normalization-completion}
Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring.
Let $X \to \Spec(A)$ be a morphism which is locally of finite type.
Set $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$.
If the complement of the special fibre in $Y$ is normal, then
the normalization $X^\nu \to X$ is finite and the base change
of $X^\nu$ to $\Spec(A^\wedge)$ recovers the normalization of $Y$.
\end{lemma}
\begin{proof}
There is an immediate reduction to the case where $X = \Spec(B)$
is affine with $B$ a finite type $A$-algebra. Set $C = B \otimes_A A^\wedge$
so that $Y = \Spec(C)$. Since
$A \to A^\wedge$ is faithfully flat, for any prime $\mathfrak q \subset B$
there exists a prime $\mathfrak r \subset C$ lying over $\mathfrak q$.
Then $B_\mathfrak q \to C_\mathfrak r$ is faithfully flat. Hence if
$\mathfrak q$ does not lie over $\mathfrak m$, then $C_\mathfrak r$
is normal by assumption on $Y$ and we conclude that $B_\mathfrak q$
is normal by Algebra, Lemma \ref{algebra-lemma-descent-normal}.
In this way we see that $X$ is normal away from the special fibre.
\medskip\noindent
Recall that the complete Noetherian local ring $A^\wedge$ is Nagata
(Algebra, Lemma \ref{algebra-lemma-Noetherian-complete-local-Nagata}).
Hence the normalization $Y^\nu \to Y$ is finite
(Morphisms, Lemma \ref{morphisms-lemma-nagata-normalization})
and an isomorphism away from the special fibre. Say $Y^\nu = \Spec(C')$.
Then $C \to C'$ is finite and an isomorphism away from $V(\mathfrak m C)$.
Since $B \to C$ is flat and induces an isomorphism
$B/\mathfrak m B \to C/\mathfrak m C$ there exists a finite
ring map $B \to B'$ whose base change to $C$ recovers $C \to C'$.
See More on Algebra, Lemma
\ref{more-algebra-lemma-application-formal-glueing} and
Remark \ref{more-algebra-remark-formal-glueing-algebras}.
Thus we find a finite morphism $X' \to X$ which is an isomorphism
away from the special fibre and whose base change recovers $Y^\nu \to Y$.
By the discussion in the first paragraph we see that $X'$ is normal at
points not on the special fibre. For a point $x \in X'$ on the special
fibre we have a corresponding point $y \in Y^\nu$ and a flat map
$\mathcal{O}_{X', x} \to \mathcal{O}_{Y^\nu, y}$.
Since $\mathcal{O}_{Y^\nu, y}$ is normal, so is $\mathcal{O}_{X', x}$, see
Algebra, Lemma \ref{algebra-lemma-descent-normal}.
Thus $X'$ is normal and it follows that it is the normalization of $X$.
\end{proof}
\begin{lemma}
\label{lemma-normalized-blowup-completion}
Let $(A, \mathfrak m, \kappa)$ be a Noetherian local domain whose completion
$A^\wedge$ is normal. Then given any sequence
$$
Y_n \to Y_{n - 1} \to \ldots \to Y_1 \to \Spec(A^\wedge)
$$
of normalized blowups, there exists a sequence of (proper) normalized blowups
$$
X_n \to X_{n - 1} \to \ldots \to X_1 \to \Spec(A)
$$
whose base change to $A^\wedge$ recovers the given sequence.
\end{lemma}
\begin{proof}
Given the sequence $Y_n \to \ldots \to Y_1 \to Y_0 = \Spec(A)^\wedge)$ we
inductively construct $X_n \to \ldots \to X_1 \to X_0 = \Spec(A)$.
The base case is $i = 0$. Given $X_i$ whose base change is $Y_i$,
let $Y'_i \to Y_i$ be the blowing up in the closed point $y_i \in Y_i$
such that $Y_{i + 1}$ is the normalization of $Y_i$.
Since the closed fibres of $Y_i$ and $X_i$ are isomorphic, the point
$y_i$ corresponds to a closed point $x_i$ on the special fibre of $X_i$.
Let $X'_i \to X_i$ be the blowup of $X_i$ in $x_i$. Then the base change
of $X'_i$ to $\Spec(A^\wedge)$ is isomorphic to $Y'_i$.
By Lemma \ref{lemma-normalization-completion}
the normalization $X_{i + 1} \to X'_i$ is finite and its base change
to $\Spec(A^\wedge)$ is isomorphic to $Y_{i + 1}$.
\end{proof}
\section{Rational double points}
\label{section-rational-double-points}
\noindent
In Section \ref{section-rational-singularities}
we argued that resolution of $2$-dimensional
rational singularities reduces to the Gorenstein case.
A Gorenstein rational surface singularity is a rational double point.
We will resolve them by explicit computations.
\medskip\noindent
According to the discussion in Examples, Section \ref{examples-section-bad}
there exists a normal Noetherian local domain $A$ whose completion
is isomorphic to $\mathbf{C}[[x, y, z]]/(z^2)$. In this case one could
say that $A$ has a rational double point singularity, but on the other
hand, $\Spec(A)$ does not have a resolution of singularities.
This kind of behaviour cannot occur if $A$ is a Nagata ring, see
Algebra, Lemma \ref{algebra-lemma-local-nagata-domain-analytically-unramified}.
\medskip\noindent
However, it gets worse as there exists a local normal Nagata domain $A$
whose completion is $\mathbf{C}[[x, y, z]]/(yz)$ and another whose
completion is $\mathbf{C}[[x, y, z]]/(y^2 - z^3)$. This is Example 2.5 of
\cite{Nishimura-few}. This is why we need to assume the completion of
our ring is normal in this section.
\begin{situation}
\label{situation-rational-double-point}
Here $(A, \mathfrak m, \kappa)$ be a Nagata local normal domain of
dimension $2$ which defines a rational singularity, whose completion
is normal, and which is Gorenstein. We assume $A$ is not regular.
\end{situation}
\noindent
The arguments in this section will show that repeatedly blowing
up singular points resolves $\Spec(A)$ in this situation. We will
need the following lemma in the course of the proof.
\begin{lemma}
\label{lemma-issquare}
Let $\kappa$ be a field. Let $I \subset \kappa[x, y]$ be an ideal. Let
$$
a + b x + c y + d x^2 + exy + f y^2 \in I^2
$$
for some $a, b, c, d, e, f \in k$ not all zero. If the colength
of $I$ in $\kappa[x, y]$ is $> 1$, then
$a + b x + c y + d x^2 + exy + f y^2 = j(g + hx + iy)^2$
for some $j, g, h, i \in \kappa$.
\end{lemma}
\begin{proof}
Consider the partial derivatives $b + 2dx + ey$ and
$c + ex + 2fy$. By the Leibniz rules these are contained in $I$.
If one of these is nonzero, then after a linear change of coordinates,
i.e., of the form $x \mapsto \alpha + \beta x + \gamma y$ and
$y \mapsto \delta + \epsilon x + \zeta y$, we may assume
that $x \in I$. Then we see that $I = (x)$ or $I = (x, F)$ with
$F$ a monic polynomial of degree $\geq 2$ in $y$.
In the first case the statement is clear. In the second case
observe that we can can write any element in $I^2$ in the form
$$
A(x, y) x^2 + B(y) x F + C(y) F^2
$$
for some $A(x, y) \in \kappa[x, y]$ and $B, C \in \kappa[y]$.
Thus
$$
a + b x + c y + d x^2 + exy + f y^2 = A(x, y) x^2 + B(y) x F + C(y) F^2
$$
and by degree reasons we see that $B = C = 0$ and $A$ is a constant.
\medskip\noindent
To finish the proof we need to deal with the case that both
partial derivatives are zero. This can only happen in characteristic $2$
and then we get
$$
a + d x^2 + f y^2 \in I^2
$$
We may assume $f$ is nonzero (if not, then switch the roles of $x$ and $y$).
After dividing by $f$ we obtain the case where the characteristic of
$\kappa$ is $2$ and
$$
a + d x^2 + y^2 \in I^2
$$
If $a$ and $d$ are squares in $\kappa$, then we are done. If not,
then there exists a derivation $\theta : \kappa \to \kappa$ with
$\theta(a) \not = 0$ or $\theta(d) \not = 0$, see
Algebra, Lemma \ref{algebra-lemma-derivative-zero-pth-power}.
We can extend this to a derivation of $\kappa[x, y]$ by setting
$\theta(x) = \theta(y) = 0$. Then we find that
$$
\theta(a) + \theta(d) x^2 \in I
$$
The case $\theta(d) = 0$ is absurd. Thus we may assume
that $\alpha + x^2 \in I$ for some $\alpha \in \kappa$.
Combining with the above we find that $a + \alpha d + y^2 \in I$.
Hence
$$
J = (\alpha + x^2, a + \alpha d + y^2) \subset I
$$
with codimension at most $2$. Observe that
$J/J^2$ is free over $\kappa[x, y]/J$ with basis
$\alpha + x^2$ and $a + \alpha d + y^2$.
Thus $a + d x^2 + y^2 =
1 \cdot (a + \alpha d + y^2) + d \cdot (\alpha + x^2) \in I^2$
implies that the inclusion $J \subset I$ is strict.
Thus we find a nonzero element of the form $g + hx + iy + jxy$ in $I$.
If $j = 0$, then $I$ contains a linear form and we can
conclude as in the first paragraph. Thus $j \not = 0$
and $\dim_\kappa(I/J) = 1$ (otherwise we could find
an element as above in $I$ with $j = 0$).
We conclude that $I$ has the form
$(\alpha + x^2, \beta + y^2, g + hx + iy + jxy)$
with $j \not = 0$ and has colength $3$.
In this case $a + dx^2 + y^2 \in I^2$ is impossible.
This can be shown by a direct computation, but we prefer to argue
as follows. Namely, to prove this statement we may assume that
$\kappa$ is algebraically closed. Then we can do a coordinate
change $x \mapsto \sqrt{\alpha} + x$ and $y \mapsto \sqrt{\beta} + y$
and assume that $I = (x^2, y^2, g' + h'x + i'y + jxy)$ with the same $j$.
Then $g' = h' = i' = 0$ otherwise the colength of $I$ is not $3$.
Thus we get $I = (x^2, y^2, xy)$ and the result is clear.
\end{proof}
\noindent
Let $(A, \mathfrak m, \kappa)$ be as in
Situation \ref{situation-rational-double-point}.
Let $X \to \Spec(A)$ be the blowing up of $\mathfrak m$ in $\Spec(A)$.
By Lemma \ref{lemma-blow-up-normal-rational} we see that $X$ is normal.
All singularities of $X$ are rational singularities
by Lemma \ref{lemma-rational-propagates}.
Since $\omega_A = A$ we see from Lemma \ref{lemma-dualizing-blow-up-rational}
that $\omega_X \cong \mathcal{O}_X$ (see discussion in
Remark \ref{remark-dualizing-setup} for conventions).
Thus all singularities of $X$ are Gorenstein.
Moreover, the local rings of $X$ at closed point have
normal completions by Lemma \ref{lemma-blowup-still-good}.
In other words, by blowing up $\Spec(A)$ we obtain a normal
surface $X$ whose singular points are as in
Situation \ref{situation-rational-double-point}.
We will use this below without further mention.
(Note: we will see in the course of the discussion below
that there are finitely many of these singular points.)
\medskip\noindent
Let $E \subset X$ be the exceptional divisor. We have
$\omega_E = \mathcal{O}_E(-1)$ by Lemma \ref{lemma-dualizing-blow-up-rational}.
By Lemma \ref{lemma-cohomology-blow-up-rational} we have
$\kappa = H^0(E, \mathcal{O}_E)$.
Thus $E$ is a Gorenstein curve and by Riemann-Roch as discussed in
Algebraic Curves, Section \ref{curves-section-Riemann-Roch}
we have
$$
\chi(E, \mathcal{O}_E) = 1 - g = -(1/2) \deg(\omega_E) =
(1/2)\deg(\mathcal{O}_E(1))
$$
where $g = \dim_\kappa H^1(E, \mathcal{O}_E) \geq 0$.
Since $\deg(\mathcal{O}_E(1))$ is positive
by Varieties, Lemma
\ref{varieties-lemma-ampleness-in-terms-of-degrees-components}
we find that $g = 0$ and $\deg(\mathcal{O}_E(1)) = 2$. It follows that
we have
$$
\dim_\kappa (\mathfrak m^n/\mathfrak m^{n + 1}) = 2n + 1
$$
by Lemma \ref{lemma-cohomology-blow-up-rational} and Riemann-Roch
on $E$.
\medskip\noindent
Choose $x_1, x_2, x_3 \in \mathfrak m$ which map to a basis of
$\mathfrak m/\mathfrak m^2$. Because
$\dim_\kappa(\mathfrak m^2/\mathfrak m^3) = 5$
the images of $x_i x_j$, $i \geq j$ in this $\kappa$-vector space
satisfy a relation. In other words, we can find $a_{ij} \in A$,
$i \geq j$, not all contained in $\mathfrak m$, such that
$$
a_{11} x_1^2 + a_{12} x_1x_2 + a_{13}x_1x_3 + a_{22} x_2^2 +
a_{23} x_2x_3 + a_{33} x_3^2 =
\sum a_{ijk} x_ix_jx_k
$$
for some $a_{ijk} \in A$ where $i \leq j \leq k$. Denote
$a \mapsto \overline{a}$ the map $A \to \kappa$.