Lemma 93.17.7. Let k be a field and let X be a scheme over k. Assume
X is separated, finite type over k and \dim (X) \leq 1,
X is a local complete intersection over k, and
X \to \mathop{\mathrm{Spec}}(k) is smooth except at finitely many points.
Then there exists a flat, separated, finite type morphism Y \to \mathop{\mathrm{Spec}}(k[[t]]) whose generic fibre is smooth and whose special fibre is isomorphic to X.
Proof.
If X is reduced, then we can choose an embedding X \subset \overline{X} as in Varieties, Lemma 33.43.6. Writing X = \overline{X} \setminus \{ x_1, \ldots , x_ n\} we see that \mathcal{O}_{\overline{X}, x_ i} is a discrete valuation ring and hence in particular a local complete intersection (Algebra, Definition 10.135.5). Thus \overline{X} is a local complete intersection over k because this holds over the open X and at the points x_ i by Algebra, Lemma 10.135.7. Thus we may apply Lemma 93.17.6 to find a projective flat morphism \overline{Y} \to \mathop{\mathrm{Spec}}(k[[t]]) whose generic fibre is smooth and whose special fibre is \overline{X}. Then we remove x_1, \ldots , x_ n from \overline{Y} to obtain Y.
In the general case, write X = X' \amalg X'' where with \dim (X') = 0 and X'' equidimensional of dimension 1. Then X'' is reduced and the first paragraph applies to it. On the other hand, X' can be dealt with as in the proof of Lemma 93.17.6. Some details omitted.
\square
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