The Stacks project

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

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

  2. 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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