The Stacks project

Lemma 54.8.5. 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

\[ X = X_ n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \mathop{\mathrm{Spec}}(A) \]

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 $X \to \mathop{\mathrm{Spec}}(A)$ is a modification and $X$ is a normal scheme projective over $A$.

Proof. We choose a modification $X \to \mathop{\mathrm{Spec}}(A)$ with $X$ normal which maximizes the length of $H^1(X, \mathcal{O}_ X)$. By Lemma 54.8.1 for any further modification $g : X' \to X$ with $X'$ normal we have $R^1g_*\mathcal{O}_{X'} = 0$ and $H^1(X, \mathcal{O}_ X) = H^1(X', \mathcal{O}_{X'})$.

Let $x \in X$ be a closed point. We will show that $\mathcal{O}_{X, x}$ defines a rational singularity. Let $Y \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ be a modification with $Y$ normal. We have to show that $H^1(Y, \mathcal{O}_ Y) = 0$. By Limits, Lemma 32.19.1 we can find a morphism of schemes $g : X' \to X$ which is an isomorphism over $X \setminus \{ x\} $ such that $X' \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ is isomorphic to $Y$. Then $g$ is a modification as it is proper by Limits, Lemma 32.19.2. 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.

The conclusion is that we've found one normal modification $X$ of $\mathop{\mathrm{Spec}}(A)$ such that the local rings of $X$ at closed points all define rational singularities. Then we choose a sequence of normalized blowups $X_ n \to \ldots \to X_1 \to \mathop{\mathrm{Spec}}(A)$ such that $X_ n$ dominates $X$, see Lemma 54.5.3. For a closed point $x' \in X_ n$ mapping to $x \in X$ we can apply Lemma 54.8.4 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. $\square$


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