The Stacks project

[Corollary 2.7, DM]

Theorem 107.24.3. Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$ and genus $g \geq 2$. Then

  1. there exists an extension of discrete valuation rings $R \subset R'$ inducing a finite separable extension of fraction fields $K'/K$ and a stable family of curves $Y \to \mathop{\mathrm{Spec}}(R')$ of genus $g$ with $Y_{K'} \cong C_{K'}$ over $K'$, and

  2. there exists a finite separable extension $L/K$ and a stable family of curves $Y \to \mathop{\mathrm{Spec}}(A)$ of genus $g$ where $A \subset L$ is the integral closure of $R$ in $L$ such that $Y_ L \cong C_ L$ over $L$.

Proof. Part (1) is an immediate consequence of Lemma 107.24.1 and Semistable Reduction, Theorem 55.18.1.

Proof of (2). Let $L/K$ be the finite separable extension found in part (3) of Semistable Reduction, Theorem 55.18.1. Let $A \subset L$ be the integral closure of $R$. Recall that $A$ is a Dedekind domain finite over $R$ with finitely many maximal ideals $\mathfrak m_1, \ldots , \mathfrak m_ n$, see More on Algebra, Remark 15.102.6. Set $S = \mathop{\mathrm{Spec}}(A)$, $S_ i = \mathop{\mathrm{Spec}}(A_{\mathfrak m_ i})$, $U = \mathop{\mathrm{Spec}}(L)$, and $U_ i = S_ i \setminus \{ \mathfrak m_ i\} $. Observe that $U \cong U_ i$ for $i = 1, \ldots , n$. Set $X = C_ L$ viewed as a scheme over the open subscheme $U$ of $S$. By our choice of $L$ and $A$ and Lemma 107.24.1 we have a stable families of curves $X_ i \to S_ i$ and isomorphisms $X \times _ U U_ i \cong X_ i \times _{S_ i} U_ i$. By Limits of Spaces, Lemma 68.18.4 we can find a finitely presented morphism $Y \to S$ whose base change to $S_ i$ is isomorphic to $X_ i$ for $i = 1, \ldots , n$. Alternatively, you can use that $S = \bigcup _{i = 1, \ldots , n} S_ i$ is an open covering of $S$ and $S_ i \cap S_ j = U$ for $i \not= j$ and use $n - 1$ applications of Limits of Spaces, Lemma 68.18.1 to get $Y \to S$ whose base change to $S_ i$ is isomorphic to $X_ i$ for $i = 1, \ldots , n$. Clearly $Y \to S$ is the stable family of curves we were looking for. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0E98. Beware of the difference between the letter 'O' and the digit '0'.