The Stacks project

Lemma 55.9.2. Let $X$ be a regular model of a smooth curve $C$ over $K$. Then

  1. $X \to \mathop{\mathrm{Spec}}(R)$ is a Gorenstein morphism of relative dimension $1$,

  2. each of the irreducible components $C_ i$ of $X_ k$ is Gorenstein.

Proof. Since $X \to \mathop{\mathrm{Spec}}(R)$ is flat, to prove (1) it suffices to show that the fibres are Gorenstein (Duality for Schemes, Lemma 48.25.3). The generic fibre is a smooth curve, which is regular and hence Gorenstein (Duality for Schemes, Lemma 48.24.3). For the special fibre $X_ k$ we use that it is an effective Cartier divisor on a regular (hence Gorenstein) scheme and hence Gorenstein for example by Dualizing Complexes, Lemma 47.21.6. The curves $C_ i$ are Gorenstein by the same argument. $\square$

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