The Stacks project

Lemma 54.12.3. Let $(A, \mathfrak m, \kappa )$ be a local normal Nagata domain of dimension $2$ which defines a rational singularity, whose completion is normal, and which is Gorenstein. Then there exists a finite sequence of blowups in singular closed points

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

such that $X_ n$ is regular and such that each intervening schemes $X_ i$ is normal with finitely many singular points of the same type.

Proof. This is exactly what was proved in the discussion above. $\square$

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