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50.1. Introduction

This chapter discusses resolution of singularities of surfaces following Lipman [Lipman] and mostly following the exposition of Artin in [Artin-Lipman]. The main result (Theorem 50.14.5) 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.

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 15.44, 15.47, and 15.49 and More on Algebra, Lemma 15.39.2. Thus such a $Y$ has a resolution of singularities.

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

  1. N    replace $A$ by its normalization,
  2. V    prove Grauert-Riemenschneider,
  3. B    show there is a maximum $g$ of the lengths of $H^1(X, \mathcal{O}_X)$ over all normal modifications $X \to \mathop{\mathrm{Spec}}(A)$ and reduce to the case $g = 0$,
  4. 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 $g = 0$,
  5. D    we say $A$ defines a rational double point if $g = 0$ and $A$ is Gorenstein and in this case we explicitly resolve singularities.

Each of these steps needs assumptions on the ring $A$. We will discuss each of these in turn.

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 50.11.5. However, the way our proof works, it turns out it is easier to use Lemma 50.11.6 to lift finiteness of the normalization over the completion to finiteness of the normalization over $A$.

Ad V: This is Proposition 50.7.8 and it roughly states that for a normal modification $f : X \to \mathop{\mathrm{Spec}}(A)$ one has $R^1f_*\omega_X = 0$ where $\omega_X$ is the dualizing module of $X/A$ (Remark 50.7.7). In fact, by duality the result is equivalent to a statement (Lemma 50.7.6) about the object $Rf_*\mathcal{O}_X$ in the derived category $D(A)$. Having said this, the proof uses the standard fact that components of the special fibre have positive conormal sheaves (Lemma 50.7.4).

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 50.8.3. If $g$ (as defined above) is bounded, then a straightforward argument shows that we can find a normal modification $X \to \mathop{\mathrm{Spec}}(A)$ such that all singular points of $X$ are rational singularities, see Lemma 50.8.5. 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 50.8.5 and (2) if the fraction field extension has degree $p$, the characteristic is $p$, and $A$ is regular and complete, see Lemma 50.8.10.

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 50.9.4.

Ad D: The resolution of rational double points proceeds more or less by hand, see Section 50.12. 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 50.10 to demonstrate that the process stops.

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 $\mathop{\mathrm{Spec}}(A)$. In order to do this we first show that if a resolution exists, then there is a resolution by normalized blowups (Lemma 50.14.3). A sequence of normalized blowups can be lifted from the completion by Lemma 50.11.7. 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 50.14.4. With a stronger result in B (such as is proved in Lipman's paper) this step could be avoided.

    The code snippet corresponding to this tag is a part of the file resolve.tex and is located in lines 17–151 (see updates for more information).

    \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.

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