Situation 54.7.1. Here $(A, \mathfrak m, \kappa )$ be a local Noetherian normal domain of dimension $2$. Let $s$ be the closed point of $S = \mathop{\mathrm{Spec}}(A)$ and $U = S \setminus \{ s\}$. Let $f : X \to S$ be a modification. We denote $C_1, \ldots , C_ r$ the irreducible components of the special fibre $X_ s$ of $f$.

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