The Stacks project

37.50 Improving morphisms of relative dimension one

We can make any curve be smooth and projective after extending the ground field, compactifying, and normalizing. This also implies results about finite type morphisms whose generic fibres have dimension $1$.

Lemma 37.50.1. Let $f : X \to S$ be a morphism of schemes. Let $\eta \in S$ be a generic point of an irreducible component of $S$. Assume $f$ is separated, of finite presentation, and $\dim (X_\eta ) \leq 1$. Then there exists a commutative diagram

\[ \xymatrix{ \overline{Y}_1 \amalg \ldots \amalg \overline{Y}_ n \ar[rd] & Y_1 \amalg \ldots \amalg Y_ n \ar[r]_-\nu \ar[d] \ar[l]^ j & X_ V \ar[r] \ar[d] & X_ U \ar[r] \ar[d] & X \ar[d]^ f \\ & T_1 \amalg \ldots \amalg T_ n \ar[r] & V \ar[r] & U \ar[r] & S } \]

of schemes with the following properties:

  1. $U \subset X$ is an open neighbourhood of $\eta $,

  2. $V \to U$ is a finite, surjective, universally injective morphism,

  3. $X_ U = U \times _ S X$ and $X_ V = V \times _ S X$ are the base changes,

  4. $\nu $ is finite, surjective, and there is an open $W \subset X_ V$ such that

    1. $W$ is dense in all fibres of $X_ V \to V$,

    2. $\nu ^{-1}(W) \cap Y_ i$ is dense in all fibres of $Y_ i \to T_ i$, and

    3. $\nu ^{-1}(W) \to W$ is a thickening,

  5. $j$ is an open immersion,

  6. $T_ i \to V$ is finite étale,

  7. $Y_ i \to T_ i$ is surjective and smooth,

  8. $\overline{Y}_ i \to T_ i$ is smooth, proper, with geometrically connected fibres of dimension $\leq 1$.

Proof. It is clear that we may replace $S$ by an open neighbourhood of $\eta $ and $X$ by the restriction to this open. Moreover, we may replace $S$ by its reduction and $X$ by the base change to this reduction. Thus we may assume $S = \mathop{\mathrm{Spec}}(A)$ where $A$ is a reduced ring and $\eta $ corresponds to a minimal prime ideal $\mathfrak p$. Recall that the local ring $\mathcal{O}_{S, \eta } = A_\mathfrak p$ is equal to $\kappa (\mathfrak p)$ in this case, see Algebra, Lemma 10.25.1.

Apply Varieties, Lemma 33.42.7 to the scheme $X_\eta $ over $k = \kappa (\eta )$. Denote $k'/k$ the purely inseparable field extension this produces. In the next paragraph we reduce to the case $k' = k$. (This step corresponds to finding the morphism $V \to U$ in the statement of the lemma; in particular we can take $V = U$ if the characteristic of $\kappa (\mathfrak p)$ is zero.)

If the characteristic of $k = \kappa (\mathfrak p)$ is zero, then $k' = k$. If the characteristic of $k = \kappa (\mathfrak p)$ is $p > 0$, then $p$ maps to zero in $A_\mathfrak p = \kappa (\mathfrak p)$. Hence after replacing $A$ by a principal localization (i.e., shrinking $S$) we may assume $p = 0$ in $A$. If $k' \not= k$, then there exists an $\beta \in k'$, $\beta \not\in k$ such that $\beta ^ p \in k$. After replacing $A$ by a principal localization we may assume there exists an $a \in A$ such that $\beta ^ p = a$. Set $A' = A[x]/(x^ p - a)$. Then $S' = \mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(A) = S$ is finite, surjective, and universally injective. Moreover, if $\mathfrak p' \subset A'$ denotes the unique prime ideal lying over $\mathfrak p$, then $A'_{\mathfrak p'} = k(\beta )$ and $k'/k(\beta )$ has smaller degree. Thus after replacing $S$ by $S'$ and $\eta $ by the point $\eta '$ corresponding to $\mathfrak p'$ we see that the degree of $k'$ over the residue field of $\eta $ has decreased. Continuing like this, by induction we reduce to the case $k' = \kappa (\mathfrak p) = \kappa (\eta )$.

Thus we may assume $S$ is affine, reduced, and that we have a diagram

\[ \xymatrix{ \overline{Y}_{1, \eta } \amalg \ldots \amalg \overline{Y}_{n, \eta } \ar[rd] & Y_{1, \eta } \amalg \ldots \amalg Y_{n, \eta } \ar[r]_-\nu \ar[d] \ar[l]^ j & X_\eta \ar[d] \\ & \mathop{\mathrm{Spec}}(k_1) \amalg \ldots \amalg \mathop{\mathrm{Spec}}(k_ n) \ar[r] & \eta } \]

of schemes with the following properties:

  1. $\nu $ is the normalization of $X_\eta $,

  2. $j$ is an open immersion with dense image,

  3. $k_ i/\kappa (\eta )$ is a finite separable extension for $i = 1, \ldots , n$,

  4. $\overline{Y}_{i, \eta }$ is smooth, projective, and geometrically irreducible of dimension $\leq 1$ over $k_ i$.

Recall that $\kappa (\eta ) = \kappa (\mathfrak p) = A_\mathfrak p$ is the filtered colimit of $A_ a$ for $a \in A$, $a \not\in \mathfrak p$. See Algebra, Lemma 10.9.9. Thus we can descend the diagram above to a corresponding diagram over $\mathop{\mathrm{Spec}}(A_ a)$ for some $a \in A$, $a \not\in \mathfrak p$. More precisely, after replacing $S$ by $\mathop{\mathrm{Spec}}(A_ a)$ we may assume we have a commutative diagram

\[ \xymatrix{ \overline{Y}_1 \amalg \ldots \amalg \overline{Y}_ n \ar[rd] & Y_1 \amalg \ldots \amalg Y_ n \ar[r]_-\nu \ar[d] \ar[l]^ j & X \ar[d] \\ & T_1 \amalg \ldots \amalg T_ n \ar[r] & S } \]

of schemes whose base change to $\eta $ is the diagram above with the following properties

  1. $\nu $ is a finite, surjective morphism,

  2. $j$ is an open immersion,

  3. $T_ i \to S$ is finite étale for $i = 1, \ldots , n$,

  4. $Y_ i \to T_ i$ is smooth and surjective,

  5. $\overline{Y}_ i \to T_ i$ is smooth and proper and has geometrically connected fibres of dimension $\leq 1$.

For this we first use Limits, Lemma 32.10.1 to obtain the diagram base changing to the previous diagram. Then we use Limits, Lemmas 32.8.10, 32.8.9, 32.8.3, 32.4.13, 32.8.12, 32.13.1, and 32.8.14 to obtain $\nu $ finite, surjective, $j$ open immersion, $T_ i \to S$ finite étale, $Y_ i \to T$ smooth, $\overline{Y}_ i \to T_ i$ proper and smooth. Since $Y_ i$ cannot be empty, since smooth morphisms are open, and since $T_ i \to S$ is finite étale, after shrinking $S$ we may assume $Y_ i \to T_ i$ is surjective. Finally, the fibre of $\overline{Y}_ i \to T_ i$ over the unique point $\eta _ i = \mathop{\mathrm{Spec}}(k_ i)$ of $T_ i$ lying over $\eta $ is geometrically connected. Hence by another shrinking we may assume the same thing is true for all fibres, see Lemma 37.49.8.

It remains to prove the existence of an open $W \subset X$ satisfying (a), (b), and (c). Since $\nu _\eta : \coprod Y_{i, \eta } \to X_\eta $ is the normalization morphism, we know by Varieties, Lemma 33.27.1 there exists a dense open $W_\eta \subset X_\eta $ such that $\nu ^{-1}(W_\eta ) \to W_\eta $ is equal to the inclusion of the reduction of $W_\eta $ into $W_\eta $. Let $W \subset X$ be a quasi-compact open whose fibre over $\eta $ is the open $W_\eta $ we just found. After replacing $A = \Gamma (S, \mathcal{O}_ S)$ by another localization we may assume $\nu ^{-1}(W) \to W$ is a closed immersion, see Limits, Lemma 32.8.5. Since $\nu $ is also surjective we conclude $\nu ^{-1}(W) \to W$ is a thickening. Set $W_ i = \nu ^{-1}(W) \cap Y_ i$. Shrinking $S$ once more we can assume $W_ i \to T_ i$ is surjective for all $i$ (same argument as above). Then we find that $W_ i \subset Y_ i$ is dense in all fibres of $Y_ i \to T_ i$ as $Y_ i \to T_ i$ has geometrically irreducible fibres. Since $\nu $ is finite and surjective, it then follows that $W = \nu (\nu ^{-1}(W))$ is dense in all fibres of $X \to S$ too. $\square$


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