The Stacks project

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 $U \to W$ ├ętale 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 37.32.2 we can choose an ├ętale 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 37.35.3 to conclude. $\square$


Comments (2)

Comment #5115 by Tongmu He on

It seems that in the statement of 53.20.12, we could add "the image of vanishes at the base point of , and the base point of maps to the node of the fiber ", right?

Comment #5322 by on

Dear Tongmu He, we don't need to say this because it'll automatically be true. Namely, the point which maps to must be a singularity of its fibre. Hence it must be true that the image of in the spectrum of is in the locus of singular points of fibres of the map to the spectrum of . (It is rather terrible that the coordinates are called but I'm going to leave this for now.)


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