Lemma 53.20.1. Let $f : X \to S$ be a morphism of schemes. The following are equivalent
$f$ is flat, locally of finite presentation, every nonempty fibre $X_ s$ is equidimensional of dimension $1$, and $X_ s$ has at-worst-nodal singularities, and
$f$ is syntomic of relative dimension $1$ and the closed subscheme $\text{Sing}(f) \subset X$ defined by the first Fitting ideal of $\Omega _{X/S}$ is unramified over $S$.
Proof. Recall that the formation of $\text{Sing}(f)$ commutes with base change, see Divisors, Lemma 31.10.1. Thus the lemma follows from Lemma 53.19.15, Morphisms, Lemma 29.30.11, and Morphisms, Lemma 29.35.12. (We also use the trivial Morphisms, Lemmas 29.30.6 and 29.30.7.) $\square$
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