Definition 76.55.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is at-worst-nodal of relative dimension $1$ if the equivalent conditions of Morphisms of Spaces, Lemma 67.22.1 hold with $\mathcal{P} =$“at-worst-nodal of relative dimension $1$”.
76.55 Families of nodal curves
This section is the continuation of Algebraic Curves, Section 53.20. Please also see that section for our choice of terminology.
The property “at-worst-nodal of relative dimension $1$” of morphisms of schemes is étale local on the source-and-target, see Descent, Lemma 35.32.6 and Algebraic Curves, Lemmas 53.20.8, 53.20.9, and 53.20.7. It is also stable under base change and fpqc local on the target, see Algebraic Curves, Lemmas 53.20.4 and 53.20.9. Hence, by Morphisms of Spaces, Lemma 67.22.1 we may define the notion of an at-worst-nodal morphism of relative dimension $1$ for algebraic spaces as follows and it agrees with the already existing notion defined in Morphisms of Spaces, Section 67.3 when the morphism is representable.
Lemma 76.55.2. The property of being at-worst-nodal of relative dimension $1$ is preserved under base change.
Proof. See Morphisms of Spaces, Remark 67.22.4 and Algebraic Curves, Lemma 53.20.4. $\square$
Lemma 76.55.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:
$f$ is at-worst-nodal of relative dimension $1$,
for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is at-worst-nodal of relative dimension $1$,
for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is at-worst-nodal of relative dimension $1$,
there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $V \times _ Y X \to V$ is at-worst-nodal of relative dimension $1$,
there exists a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi $ is at-worst-nodal of relative dimension $1$,
for every commutative diagram
where $U$, $V$ are schemes and the vertical arrows are étale the top horizontal arrow is at-worst-nodal of relative dimension $1$,
there exists a commutative diagram
where $U$, $V$ are schemes, the vertical arrows are étale, and $U \to X$ is surjective such that the top horizontal arrow is at-worst-nodal of relative dimension $1$, and
there exist Zariski coverings $Y = \bigcup _{i \in I} Y_ i$, and $f^{-1}(Y_ i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_ i$ is at-worst-nodal of relative dimension $1$.
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]
Proof. Omitted. $\square$
The following lemma tells us that we can check whether a morphism is at-worst-nodal of relative dimension $1$ on the fibres.
Lemma 76.55.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Then there is a maximal open subspace $X' \subset X$ such that $f|_{X'} : X' \to Y$ is at-worst-nodal of relative dimension $1$. Moreover, formation of $X'$ commutes with arbitrary base change.
Proof. Choose a commutative diagram
\[ \xymatrix{ U \ar[d] \ar[r]_ h & V \ar[d] \\ X \ar[r]^ f & Y } \]
where $U$, $V$ are schemes, the vertical arrows are étale, and $U \to X$ is surjective. By the lemma for the case of schemes (Algebraic Curves, Lemma 53.20.5) we find a maximal open subscheme $U' \subset U$ such that $h|_{U'} : U' \to V$ is at-worst-nodal of relative dimension $1$ and such that formation of $U'$ commutes with base change. Let $X' \subset X$ be the open subspace whose points correspond to the open subset $\mathop{\mathrm{Im}}(|U'| \to |X|)$. By Lemma 76.55.3 we see that $X' \to Y$ is at-worst-nodal of relative dimension $1$ and that $X'$ is the largest open subspace with this property (this also implies that $U'$ is the inverse image of $X'$ in $U$, but we do not need this). Since the same is true after base change the proof is complete. $\square$
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