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. )



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


