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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