# Does there exist a genus $g$ curve over $\mathbb{Q}$ with every type of stable reduction?

Let $$g\geq3$$ be an integer, let $$\{\Gamma_i|i \in I\}$$ be the set of all stable graphs of genus $$g$$. (We say a graph is stable if it is the dual graph of a stable curve.)

Let $$X$$ be a curve defined over $$\mathbb{Q}$$, we say it has reduction type $$\Gamma_i$$ , if there is a model $$\mathcal{X}_{p_i}$$ over $$\mathbb{Z}_{p_i}$$, whose generic fiber is $$X_{\mathbb{Q}_{p_i}}$$ and special fiber is a stable curve with dual graph $$\Gamma_i$$.

Given $$g\geq3$$, does there always exist a smooth curve over $$X/\mathbb{Q}$$, (or some number field $$K$$?) such that every $$\Gamma_i$$ appear as the reduction of certain prime $$p_i$$?

(If we ask the same question over global function field (finite extension of $$\mathbb{F}_p(t)$$), I think the answer is "yes", as we can interpolate the loci in $$\overline{\mathcal{M}}_g$$ by curves. )

• A smooth curve over $\mathbb Q$ only has finitely many points of bad reduction. – Angelo Oct 12 '19 at 6:48
• @Angelo Yes, but there are only finitely many stable graphs in fixed genus. – Olivier Benoist Oct 12 '19 at 10:10
• Just to clarify, is the question asking for every $g \geq 3$ there exists such a curve, or just for some $g$ or possibly infinitely many such $g$? – Stanley Yao Xiao Oct 12 '19 at 11:59
• @StanleyYaoXiao Thanks! I've edited the question – Qixiao Oct 12 '19 at 13:00
• Oops, sorry, this was a really moronic comment. – Angelo Oct 12 '19 at 15:26

For any list $$p_i$$ of primes, we can find such a curve over a number field that has reduction type $$\Gamma_i$$ at some prime lying over $$p_i$$.

We can iteratively blow up the strata of the stable graph stratification of $$\overline{M}_g$$ until they all have codimension $$1$$. Then by taking an intersection of general hyperplanes in the blow-up and applying Bertini, we can find an irreducible curve in $$\overline{M}_g$$ that meets every locus. Now map that curve to $$\mathbb P^1$$. Take a number field $$K'$$ where the images of the meeting points with every locus are defined, and pull back the $$p_i$$ to primes $$\mathfrak p_i$$ of $$K'$$. We can easily find a point of $$\mathbb P^1(K')$$ which is congruent mod $$\mathfrak p_i$$ to the image of the meeting point with the $$\Gamma_i$$ stratum, as this is just finitely many congruence conditions. We can now lift this point to a point of the curve, over a possibly larger number field, and then to a point of $$\overline{\mathcal M}_g$$ with stable reduction, over an even larger number field. At the point where it meets the $$\Gamma_i$$ locus in $$\mathcal M_g$$, becaue it has stable reduction, it must in fact have reduction type $$\Gamma_i$$.

I am sure the question over $$\mathbb Q$$ is open, as not much is known about the rational points on $$M_g$$ in general, except for very small $$g$$ which are unirational.

Here is a stacky variant of Will Savin's answer, with a more precise result. The reference is my paper:

Problèmes de Skolem sur les champs algébriques, Compo. Math. 125 (2001), 1—30

(1) First, note the following: Let $$F$$ be a local field, $$Y$$ an $$F$$-scheme of finite type, $$C\to Y$$ a stable curve of genus $$g$$ over $$Y$$, and $$\Gamma$$ a stable graph of genus $$g$$. Let $$U$$ (resp. $$U'\subset U$$) be the set of points $$y\in Y(F)$$ such that $$C_y$$ has (potential) reduction type $$\Gamma$$ (resp. has stable reduction on $$O_F$$, with reduction type $$\Gamma$$). Then both $$U$$ and $$U'$$ are open in $$Y(F)$$, for the valuation topology. (Exercise).

(2) Now let $$I'\subset I$$ correspond to singular curves (i.e. we remove the one-point graph from $$I$$).
For each $$i\in I'$$, fix a prime $$p_i$$ (they have to be pairwise distinct!). I will make the assumption that
(*) there exists a smooth curve over $$\mathbb{Q}_{p_i}$$ with potential reduction graph $$\Gamma_i$$.
(I am pretty sure that this is always true; otherwise, choose $$p_i$$ accordingly).
Put $$R:=\mathbb{Z}\left[(1/p_i)_{i\in I'}\right]$$.

Theorem. There is a number field $$K$$, and a stable curve $$\mathscr{X}$$ over $$O_K$$ such that:
$$\mathscr{X}$$ is smooth over $$R\otimes_{\mathbb{Z}}O_K$$;
• for each $$i\in I'$$ and each prime $$\mathfrak{p}$$ of $$O_K$$ above $$p_i$$, $$\mathscr{X}$$ has reduction graph $$\Gamma_i$$ at $$\mathfrak{p}$$.
Assume further that for each $$i\in I'$$ there is a stable curve over $$\mathbb{F}_{p_i}$$ with graph $$\Gamma_i$$.
(This holds if $$p_i$$ is large enough). Then we can take $$K$$ totally split at each $$p_i$$.

Proof: Consider the moduli stack $$\mathscr{M}_g$$ (resp. $$\overline{\mathscr{M}}_g$$) of smooth (resp. stable) curves of genus $$g$$.
Let us prove the first claim. For each $$i$$, let $$\Omega_i\subset{\mathscr{M}}_g(\mathbb{Q}_{p_i})$$ be the subcategory of those smooth curves with (potential) reduction graph $$\Gamma_i$$. By (1), this is $$p_i$$-adically open in $${\mathscr{M}}_g(\mathbb{Q}_{p_i})$$, in the sense of Definition 2.2 in the paper. Moreover, it is not empty, due to assumption (*).
It is straightforward to check that $$\mathscr{M}_{g,R}\to\mathrm{Spec}(R)$$ with the local data $$(\Omega_i)_{i\in I'}$$ constitutes a Skolem datum in the sense of Definition 0.6. (We use the fact that $$\mathscr{M}_g$$ is smooth with geometrically connected fibers over $$\mathrm{Spec}(\mathbb{Z})$$). Now apply Theorem 0.7: this almost gives the result (with $$K$$ totally split at each $$p_i$$) except that the curve may not have stable reduction outside $$\mathrm{Spec}(R)$$. To fix this, just enlarge $$K$$, possibly losing splitness.
For the second claim, we do the same, replacing $$\Omega_i$$ by $$\Omega'_i$$ consisting of curves with stable reduction of type $$\Gamma_i$$ over $$\mathbb{Z}_{p_i}$$: the extra assumption on $$p_i$$ guarantees that $$\Omega'_i\neq\emptyset$$. QED