In the Stacks project curves are irreducible varieties of dimension $1$, but in the literature a “semi-stable curve” or a “nodal curve” is usually not irreducible and often assumed to be proper, especially when used in a phrase such as “family of semistable curves” or “family of nodal curves”, or “nodal family”. Thus it is a bit difficult for us to choose a terminology which is consistent with the literature as well as internally consistent. Moreover, we really want to first study the notion introduced in the following lemma (which is local on the source).
Lemma 53.20.1. Let $f : X \to S$ be a morphism of schemes. The following are equivalent
$f$ is flat, locally of finite presentation, every nonempty fibre $X_ s$ is equidimensional of dimension $1$, and $X_ s$ has at-worst-nodal singularities, and
$f$ is syntomic of relative dimension $1$ and the closed subscheme $\text{Sing}(f) \subset X$ defined by the first Fitting ideal of $\Omega _{X/S}$ is unramified over $S$.
Proof. Recall that the formation of $\text{Sing}(f)$ commutes with base change, see Divisors, Lemma 31.10.1. Thus the lemma follows from Lemma 53.19.15, Morphisms, Lemma 29.30.11, and Morphisms, Lemma 29.35.12. (We also use the trivial Morphisms, Lemmas 29.30.6 and 29.30.7.) $\square$
Definition 53.20.2. Let $f : X \to S$ be a morphism of schemes. We say $f$ is at-worst-nodal of relative dimension $1$ if $f$ satisfies the equivalent conditions of Lemma 53.20.1.
Here are some reasons for the cumbersome terminology1. First, we want to make sure this notion is not confused with any of the other notions in the literature (see introduction to this section). Second, we can imagine several generalizations of this notion to morphisms of higher relative dimension (for example, one can ask for morphisms which are étale locally compositions of at-worst-nodal morphisms or one can ask for morphisms whose fibres are higher dimensional but have at worst ordinary double points).
Lemma 53.20.3. A smooth morphism of relative dimension $1$ is at-worst-nodal of relative dimension $1$.
Proof. Omitted. $\square$
Lemma 53.20.4. Let $f : X \to S$ be at-worst-nodal of relative dimension $1$. Then the same is true for any base change of $f$.
Proof. This is true because the base change of a syntomic morphism is syntomic (Morphisms, Lemma 29.30.4), the base change of a morphism of relative dimension $1$ has relative dimension $1$ (Morphisms, Lemma 29.29.2), the formation of $\text{Sing}(f)$ commutes with base change (Divisors, Lemma 31.10.1), and the base change of an unramified morphism is unramified (Morphisms, Lemma 29.35.5). $\square$
The following lemma tells us that we can check whether a morphism is at-worst-nodal of relative dimension $1$ on the fibres.
Lemma 53.20.5. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Then there is a maximal open subscheme $U \subset X$ such that $f|_ U : U \to S$ is at-worst-nodal of relative dimension $1$. Moreover, formation of $U$ commutes with arbitrary base change.
Proof. By Morphisms, Lemma 29.30.12 we find that there is such an open where $f$ is syntomic. Hence we may assume that $f$ is a syntomic morphism. In particular $f$ is a Cohen-Macaulay morphism (Duality for Schemes, Lemmas 48.25.5 and 48.25.4). Thus $X$ is a disjoint union of open and closed subschemes on which $f$ has given relative dimension, see Morphisms, Lemma 29.29.4. This decomposition is preserved by arbitrary base change, see Morphisms, Lemma 29.29.2. Discarding all but one piece we may assume $f$ is syntomic of relative dimension $1$. Let $\text{Sing}(f) \subset X$ be the closed subscheem defined by the first fitting ideal of $\Omega _{X/S}$. There is a maximal open subscheme $W \subset \text{Sing}(f)$ such that $W \to S$ is unramified and its formation commutes with base change (Morphisms, Lemma 29.35.15). Since also formation of $\text{Sing}(f)$ commutes with base change (Divisors, Lemma 31.10.1), we see that
is the maximal open subscheme of $X$ such that $f|_ U : U \to S$ is at-worst-nodal of relative dimension $1$ and that formation of $U$ commutes with base change. $\square$
Lemma 53.20.6. Let $f : X \to S$ be at-worst-nodal of relative dimension $1$. If $Y \to X$ is an étale morphism, then the composition $g : Y \to S$ is at-worst-nodal of relative dimension $1$.
Proof. Observe that $g$ is flat and locally of finite presentation as a composition of morphisms which are flat and locally of finite presentation (use Morphisms, Lemmas 29.36.11, 29.36.12, 29.21.3, and 29.25.6). Thus it suffices to prove the fibres have at-worst-nodal singularities. This follows from Lemma 53.19.13 (and the fact that the composition of an étale morphism and a smooth morphism is smooth by Morphisms, Lemmas 29.36.5 and 29.34.4). $\square$
Lemma 53.20.7. Let $S' \to S$ be an étale morphism of schemes. Let $f : X \to S'$ be at-worst-nodal of relative dimension $1$. Then the composition $g : X \to S$ is at-worst-nodal of relative dimension $1$.
Proof. Observe that $g$ is flat and locally of finite presentation as a composition of morphisms which are flat and locally of finite presentation (use Morphisms, Lemmas 29.36.11, 29.36.12, 29.21.3, and 29.25.6). Thus it suffices to prove the fibres of $g$ have at-worst-nodal singularities. This follows from Lemma 53.19.14 and the analogous result for smooth points. $\square$
Lemma 53.20.8. Let $f : X \to S$ be a morphism of schemes. Let $\{ U_ i \to X\} $ be an étale covering. The following are equivalent
$f$ is at-worst-nodal of relative dimension $1$,
each $U_ i \to S$ is at-worst-nodal of relative dimension $1$.
In other words, being at-worst-nodal of relative dimension $1$ is étale local on the source.
Proof. One direction we have seen in Lemma 53.20.6. For the other direction, observe that being locally of finite presentation, flat, or to have relative dimension $1$ is étale local on the source (Descent, Lemmas 35.28.1, 35.27.1, and 35.33.8). Taking fibres we reduce to the case where $S$ is the spectrum of a field. In this case the result follows from Lemma 53.19.13 (and the fact that being smooth is étale local on the source by Descent, Lemma 35.30.1). $\square$
Lemma 53.20.9. Let $f : X \to S$ be a morphism of schemes. Let $\{ U_ i \to S\} $ be an fpqc covering. The following are equivalent
$f$ is at-worst-nodal of relative dimension $1$,
each $X \times _ S U_ i \to U_ i$ is at-worst-nodal of relative dimension $1$.
In other words, being at-worst-nodal of relative dimension $1$ is fpqc local on the target.
Proof. One direction we have seen in Lemma 53.20.4. For the other direction, observe that being locally of finite presentation, flat, or to have relative dimension $1$ is fpqc local on the target (Descent, Lemmas 35.23.11, 35.23.15, and Morphisms, Lemma 29.28.3). Taking fibres we reduce to the case where $S$ is the spectrum of a field. In this case the result follows from Lemma 53.19.12 (and the fact that being smooth is fpqc local on the target by Descent, Lemma 35.23.27). $\square$
Lemma 53.20.10. Let $S = \mathop{\mathrm{lim}}\nolimits S_ i$ be a limit of a directed system of schemes with affine transition morphisms. Let $0 \in I$ and let $f_0 : X_0 \to Y_0$ be a morphism of schemes over $S_0$. Assume $S_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated. Let $f_ i : X_ i \to Y_ i$ be the base change of $f_0$ to $S_ i$ and let $f : X \to Y$ be the base change of $f_0$ to $S$. If
$f$ is at-worst-nodal of relative dimension $1$, and
$f_0$ is locally of finite presentation,
then there exists an $i \geq 0$ such that $f_ i$ is at-worst-nodal of relative dimension $1$.
Proof. By Limits, Lemma 32.8.16 there exists an $i$ such that $f_ i$ is syntomic. Then $X_ i = \coprod _{d \geq 0} X_{i, d}$ is a disjoint union of open and closed subschemes such that $X_{i, d} \to Y_ i$ has relative dimension $d$, see Morphisms, Lemma 29.30.14. Because of the behaviour of dimensions of fibres under base change given in Morphisms, Lemma 29.28.3 we see that $X \to X_ i$ maps into $X_{i, 1}$. Then there exists an $i' \geq i$ such that $X_{i'} \to X_ i$ maps into $X_{i, 1}$, see Limits, Lemma 32.4.10. Thus $f_{i'} : X_{i'} \to Y_{i'}$ is syntomic of relative dimension $1$ (by Morphisms, Lemma 29.28.3 again). Consider the morphism $\text{Sing}(f_{i'}) \to Y_{i'}$. We know that the base change to $Y$ is an unramified morphism. Hence by Limits, Lemma 32.8.4 we see that after increasing $i'$ the morphism $\text{Sing}(f_{i'}) \to Y_{i'}$ becomes unramified. This finishes the proof. $\square$
Lemma 53.20.11. Let $f : T \to S$ be a morphism of schemes. Let $t \in T$ with image $s \in S$. Assume
$f$ is flat at $t$,
$\mathcal{O}_{S, s}$ is Noetherian,
$f$ is locally of finite type,
$t$ is a split node of the fibre $T_ s$.
Then there exists an $h \in \mathfrak m_ s^\wedge $ and an isomorphism
of $\mathcal{O}_{S, s}^\wedge $-algebras.
Proof. We replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ and $T$ by the base change to $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$. Then $T$ is locally Noetherian and hence $\mathcal{O}_{T, t}$ is Noetherian. Set $A = \mathcal{O}_{S, s}^\wedge $, $\mathfrak m = \mathfrak m_ A$, and $B = \mathcal{O}_{T, t}^\wedge $. By More on Algebra, Lemma 15.43.8 we see that $A \to B$ is flat. Since $\mathcal{O}_{T, t}/\mathfrak m_ s \mathcal{O}_{T, t} = \mathcal{O}_{T_ s, t}$ we see that $B/\mathfrak m B = \mathcal{O}_{T_ s, t}^\wedge $. By assumption (4) and Lemma 53.19.11 we conclude there exist $\overline{u}, \overline{v} \in B/\mathfrak m B$ such that the map
is surjective with kernel $(xy)$.
Assume we have $n \geq 1$ and $u, v \in B$ mapping to $\overline{u}, \overline{v}$ such that
for some $h \in A$ and $\delta \in \mathfrak m^ nB$. We claim that there exist $u', v' \in B$ with $u - u', v - v' \in \mathfrak m^ n B$ such that
for some $h' \in A$ and $\delta ' \in \mathfrak m^{n + 1}B$. To see this, write $\delta = \sum f_ i b_ i$ with $f_ i \in \mathfrak m^ n$ and $b_ i \in B$. Then write $b_ i = a_ i + u b_{i, 1} + v b_{i, 2} + \delta _ i$ with $a_ i \in A$, $b_{i, 1}, b_{i, 2} \in B$ and $\delta _ i \in \mathfrak m B$. This is possible because the residue field of $B$ agrees with the residue field of $A$ and the images of $u$ and $v$ in $B/\mathfrak m B$ generate the maximal ideal. Then we set
and we obtain
for some $c_{i, j} \in B$. Thus we get a formula as above with $h' = h + \sum a_ if_ i$ and $\delta ' = \sum f_ i \delta _ i + \sum c_{ij}f_ if_ j$.
Arguing by induction and starting with any lifts $u_1, v_1 \in B$ of $\overline{u}, \overline{v}$ the result of the previous paragraph shows that we find a sequence of elements $u_ n, v_ n \in B$ and $h_ n \in A$ such that $u_ n - u_{n + 1} \in \mathfrak m^ n B$, $v_ n - v_{n + 1} \in \mathfrak m^ n B$, $h_ n - h_{n + 1} \in \mathfrak m^ n$, and such that $u_ n v_ n - h_ n \in \mathfrak m^ n B$. Since $A$ and $B$ are complete we can set $u_\infty = \mathop{\mathrm{lim}}\nolimits u_ n$, $v_\infty = \mathop{\mathrm{lim}}\nolimits v_ n$, and $h_\infty = \mathop{\mathrm{lim}}\nolimits h_ n$, and then we obtain $u_\infty v_\infty = h_\infty $ in $B$. Thus we have an $A$-algebra map
sending $x$ to $u_\infty $ and $v$ to $v_\infty $. This is a map of flat $A$-algebras which is an isomorphism after dividing by $\mathfrak m$. It is surjective modulo $\mathfrak m$ and hence surjective by completeness and Algebra, Lemma 10.96.1. Then we can apply Algebra, Lemma 10.99.1 to conclude it is an isomorphism. $\square$
Consider the morphism of schemes
The next lemma shows that this morphism is a model for the étale local structure of a nodal family of curves. If you know a proof of this lemma avoiding the use of Artin approximation, then please email stacks.project@gmail.com.
Lemma 53.20.12. Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is at-worst-nodal of relative dimension $1$. Let $x \in X$ be a point which is a singular point of the fibre $X_ s$. Then there exists a commutative diagram of schemes with $X \leftarrow U$, $S \leftarrow V$, and $U \to W$ étale morphisms, and with the right hand square cartesian, such that there exists a point $u \in U$ mapping to $x$ in $X$.
\[ \xymatrix{ X \ar[d] & U \ar[rr] \ar[l] \ar[rd] & & W \ar[r] \ar[ld] & \mathop{\mathrm{Spec}}(\mathbf{Z}[u, v, a]/(uv - a)) \ar[d] \\ S & & V \ar[ll] \ar[rr] & & \mathop{\mathrm{Spec}}(\mathbf{Z}[a]) } \]
Proof. We first use absolute Noetherian approximation to reduce to the case of schemes of finite type over $\mathbf{Z}$. The question is local on $X$ and $S$. Hence we may assume that $X$ and $S$ are affine. Then we can write $S = \mathop{\mathrm{Spec}}(R)$ and write $R$ as a filtered colimit $R = \mathop{\mathrm{colim}}\nolimits R_ i$ of finite type $\mathbf{Z}$-algebras. Using Limits, Lemma 32.10.1 we can find an $i$ and a morphism $f_ i : X_ i \to \mathop{\mathrm{Spec}}(R_ i)$ whose base change to $S$ is $f$. After increasing $i$ we may assume that $f_ i$ is at-worst-nodal of relative dimension $1$, see Lemma 53.20.10. The image $x_ i \in X_ i$ of $x$ will be a singular point of its fibre, for example because the formation of $\text{Sing}(f)$ commutes with base change (Divisors, Lemma 31.10.1). If we can prove the lemma for $f_ i : X_ i \to S_ i$ and $x_ i$, then the lemma follows for $f : X \to S$ by base change. Thus we reduce to the case studied in the next paragraph.
Assume $S$ is of finite type over $\mathbf{Z}$. Let $s \in S$ be the image of $x$. Recall that $\kappa (x)$ is a finite separable extension of $\kappa (s)$, for example because $\text{Sing}(f) \to S$ is unramified or because $x$ is a node of the fibre $X_ s$ and we can apply Lemma 53.19.7. Furthermore, let $\kappa '/\kappa (x)$ be the degree $2$ separable algebra associated to $\mathcal{O}_{X_ s, x}$ in Remark 53.19.8. By More on Morphisms, Lemma 37.35.2 we can choose an étale neighbourhood $(V, v) \to (S, s)$ such that the extension $\kappa (v)/\kappa (s)$ realizes either the extension $\kappa (x)/\kappa (s)$ in case $\kappa ' \cong \kappa (x) \times \kappa (x)$ or the extension $\kappa '/\kappa (s)$ if $\kappa '$ is a field. After replacing $X$ by $X \times _ S V$ and $S$ by $V$ we reduce to the situation described in the next paragraph.
Assume $S$ is of finite type over $\mathbf{Z}$ and $x \in X_ s$ is a split node, see Definition 53.19.10. By Lemma 53.20.11 we see that there exists an $\mathcal{O}_{S, s}$-algebra isomorphism
for some $h \in \mathfrak m_ s^\wedge \subset \mathcal{O}_{S, s}^\wedge $. In other words, if we consider the homomorphism
sending $a$ to $h$, then there exists an $\mathcal{O}_{S, s}$-algebra isomorphism
where
and $y_\sigma $ is the point of $Y_\sigma $ lying over the closed point of $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge )$ and having coordinates $u, v$ equal to zero. Since $\mathcal{O}_{S, s}$ is a G-ring by More on Algebra, Proposition 15.50.12 we may apply More on Morphisms, Lemma 37.39.3 to conclude. $\square$
Lemma 53.20.13. Let $f : X \to S$ be a morphism of schemes. Assume
$f$ is proper,
$f$ is at-worst-nodal of relative dimension $1$, and
the geometric fibres of $f$ are connected.
Then (a) $f_*\mathcal{O}_ X = \mathcal{O}_ S$ and this holds after any base change, (b) $R^1f_*\mathcal{O}_ X$ is a finite locally free $\mathcal{O}_ S$-module whose formation commutes with any base change, and (c) $R^ qf_*\mathcal{O}_ X = 0$ for $q \geq 2$.
Proof. Part (a) follows from Derived Categories of Schemes, Lemma 36.32.6. By Derived Categories of Schemes, Lemma 36.32.5 locally on $S$ we can write $Rf_*\mathcal{O}_ X = \mathcal{O}_ S \oplus P$ where $P$ is perfect of tor amplitude in $[1, \infty )$. Recall that formation of $Rf_*\mathcal{O}_ X$ commutes with arbitrary base change (Derived Categories of Schemes, Lemma 36.30.4). Thus for $s \in S$ we have
This is zero unless $i = 1$ since $X_ s$ is a $1$-dimensional Noetherian scheme, see Cohomology, Proposition 20.20.7. Then $P = H^1(P)[-1]$ and $H^1(P)$ is finite locally free for example by More on Algebra, Lemma 15.75.7. Since everything is compatible with base change we conclude. $\square$
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