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

$\xymatrix{ X \ar[d] & U \ar[rr] \ar[l] \ar[rd] & & W \ar[r] \ar[ld] & \mathop{\mathrm{Spec}}(\mathbf{Z}[u, v, a]/(uv - a)) \ar[d] \\ S & & V \ar[ll] \ar[rr] & & \mathop{\mathrm{Spec}}(\mathbf{Z}[a]) }$

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 $X$.

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

$\mathcal{O}_{X, x}^\wedge \cong \mathcal{O}_{S, s}^\wedge [[s, t]]/(st - h)$

for some $h \in \mathfrak m_ s^\wedge \subset \mathcal{O}_{S, s}^\wedge$. In other words, if we consider the homomorphism

$\sigma : \mathbf{Z}[a] \longrightarrow \mathcal{O}_{S, s}^\wedge$

sending $a$ to $h$, then there exists an $\mathcal{O}_{S, s}$-algebra isomorphism

$\mathcal{O}_{X, x}^\wedge \longrightarrow \mathcal{O}_{Y_\sigma , y_\sigma }^\wedge$

where

$Y_\sigma = \mathop{\mathrm{Spec}}(\mathbf{Z}[u, v, t]/(uv - a)) \times _{\mathop{\mathrm{Spec}}(\mathbf{Z}[a]), \sigma } \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^\wedge )$

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. $\square$

Comment #5115 by Tongmu He on

It seems that in the statement of 53.20.12, we could add "the image of $a$ vanishes at the base point $v$ of $V$, and the base point $u$ of $U$ maps to the node of the fiber $W_v$", right?

Comment #5322 by on

Dear Tongmu He, we don't need to say this because it'll automatically be true. Namely, the point $u \in U$ which maps to $x \in X$ must be a singularity of its fibre. Hence it must be true that the image of $u$ in the spectrum of $\mathbf{Z}[u, v, a]/(uv - a)$ is in the locus $V(u, v)$ of singular points of fibres of the map to the spectrum of $\mathbf{Z}[a]$. (It is rather terrible that the coordinates are called $u, v$ but I'm going to leave this for now.)

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