109.10 Geometrically reduced curves
There is an open substack of $\mathcal{C}\! \mathit{urves}$ parametrizing the geometrically reduced “curves”.
Lemma 109.10.1. There exist an open substack $\mathcal{C}\! \mathit{urves}^{geomred} \subset \mathcal{C}\! \mathit{urves}$ such that
given a family of curves $X \to S$ the following are equivalent
the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{geomred}$,
the fibres of the morphism $X \to S$ are geometrically reduced (More on Morphisms of Spaces, Definition 76.29.2),
given a scheme $X$ proper over a field $k$ with $\dim (X) \leq 1$ the following are equivalent
the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{geomred}$,
$X$ is geometrically reduced over $k$.
Proof.
Let $f : X \to S$ be a family of curves. By More on Morphisms of Spaces, Lemma 76.29.6 the set
\[ E = \{ s \in S : \text{the fibre of }X \to S\text{ at }s \text{ is geometrically reduced}\} \]
is open in $S$. Formation of this open commutes with arbitrary base change by More on Morphisms of Spaces, Lemma 76.29.3. Thus we get the open substack with the desired properties by the method discussed in Section 109.6.
$\square$
Lemma 109.10.2. We have $\mathcal{C}\! \mathit{urves}^{geomred} \subset \mathcal{C}\! \mathit{urves}^{CM}$ as open substacks of $\mathcal{C}\! \mathit{urves}$.
Proof.
This is true because a reduced Noetherian scheme of dimension $\leq 1$ is Cohen-Macaulay. See Algebra, Lemma 10.157.3.
$\square$
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