Definition 54.8.3. Let $(A, \mathfrak m, \kappa )$ be a local normal Nagata domain of dimension $2$.

We say $A$

*defines a rational singularity*if for every normal modification $X \to \mathop{\mathrm{Spec}}(A)$ we have $H^1(X, \mathcal{O}_ X) = 0$.We say that

*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 $X \to \mathop{\mathrm{Spec}}(A)$ with $X$ normal.

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