## 107.17 Density of smooth curves

The title of this section is misleading as we don't claim $\mathcal{C}\! \mathit{urves}^{smooth}$ is dense in $\mathcal{C}\! \mathit{urves}$. In fact, this is false as was shown by Mumford in . However, we will see that the smooth “curves” are dense in a large open.

Lemma 107.17.1. The inclusion

$|\mathcal{C}\! \mathit{urves}^{smooth}| \subset |\mathcal{C}\! \mathit{urves}^{lci+}|$

is that of an open dense subset.

Proof. By the very construction of the topology on $|\mathcal{C}\! \mathit{urves}^{lci+}|$ in Properties of Stacks, Section 98.4 we find that $|\mathcal{C}\! \mathit{urves}^{smooth}|$ is an open subset. Let $\xi \in |\mathcal{C}\! \mathit{urves}^{lci+}|$ be a point. Then there exists a field $k$ and a scheme $X$ over $k$ with $X$ proper over $k$, with $\dim (X) \leq 1$, with $X$ a local complete intersection over $k$, and with $X$ is smooth over $k$ except at finitely many points, such that $\xi$ is the equivalence class of the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}^{lci+}$ determined by $X$. See Lemma 107.15.2. By Deformation Problems, Lemma 91.17.6 there exists a flat projective morphism $Y \to \mathop{\mathrm{Spec}}(k[[t]])$ whose generic fibre is smooth and whose special fibre is isomorphic to $X$. Consider the classifying morphism

$\mathop{\mathrm{Spec}}(k[[t]]) \longrightarrow \mathcal{C}\! \mathit{urves}^{lci+}$

determined by $Y$. The image of the closed point is $\xi$ and the image of the generic point is in $|\mathcal{C}\! \mathit{urves}^{smooth}|$. Since the generic point specializes to the closed point in $|\mathop{\mathrm{Spec}}(k[[t]])|$ we conclude that $\xi$ is in the closure of $|\mathcal{C}\! \mathit{urves}^{smooth}|$ as desired. $\square$

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