Lemma 93.16.4 (0DZQ)—The Stacks project

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Table of contents
Part 6: Deformation Theory
Chapter 93: Deformation Problems
Section 93.16: Unobstructed deformation problems
Lemma 93.16.4 (cite)

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Lemma 93.16.4. In Example 93.9.1 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 $\mathcal{D}\! \mathit{ef}_ X$ is unobstructed.

Proof. Let $p_1, \ldots , p_ n \in X$ be the points where $X \to \mathop{\mathrm{Spec}}(k)$ isn't smooth. Choose an affine open $U_2 \subset X$ containing $p_1, \ldots , p_ n$ and the generic point of every irreducible component of $X$ . This is possible by Varieties, Lemma 33.43.3 and Properties, Lemma 28.29.5. Then $X \setminus U_2$ is finite and we can choose an affine open $U_1 \subset X \setminus \{ p_1, \ldots , p_ n\}$ such that $X = U_1 \cup U_2$ . Set $U_{12} = U_1 \cap U_2$ . Then $U_1$ and $U_{12}$ are smooth affine schemes over $k$ . We conclude that

$\mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_{U_2}$

is smooth by Lemma 93.16.2. Applying Lemmas 93.9.7 and 93.16.1 we win. $\square$

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