The Stacks project

107.10 Geometrically reduced curves

There is an open substack of $\mathcal{C}\! \mathit{urves}$ parametrizing the geometrically reduced “curves”.

Lemma 107.10.1. There exist an open substack $\mathcal{C}\! \mathit{urves}^{geomred} \subset \mathcal{C}\! \mathit{urves}$ such that

  1. given a family of curves $X \to S$ the following are equivalent

    1. the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{geomred}$,

    2. the fibres of the morphism $X \to S$ are geometrically reduced (More on Morphisms of Spaces, Definition 74.29.2),

  2. given a scheme $X$ proper over a field $k$ with $\dim (X) \leq 1$ the following are equivalent

    1. the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{geomred}$,

    2. $X$ is geometrically reduced over $k$.

Proof. Let $f : X \to S$ be a family of curves. By More on Morphisms of Spaces, Lemma 74.29.6 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 74.29.3. Thus we get the open substack with the desired properties by the method discussed in Section 107.6. $\square$

Lemma 107.10.2. 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 10.155.3. $\square$

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