For each genus $g$, there are many curves of genus $g$ defined over $\mathbb Q$. How many? We might study this question by considering the rational points of the Deligne-Mumford moduli space of curves $\mathcal M_g$.

Are they Zariski dense? Under the Bombieri-Lang conjecture, rational points are not conjectured to be dense in any variety of general type, and $\mathcal M_g$ is known to be dense, so probably not. So:

> What is the dimension of the largest subvariety of $\mathcal M_g$ with Zariski dense rational points?

Under the Bombieri-Lang conjecture, such a subvariety should not have any dominant rational maps to general type varieties. I think by algebraic geometry this makes it a bundle of rationally connected varieties over a variety of Kodaira dimension 0, or something like that. So one could instead ask for the largest such subvariety, a purely geometric question:

> What is the dimension of the largest subvariety of $\mathcal M_g$ that has no dominant rational maps to a variety of general type?

I'd be happy to see an conjectural and/or asymptotic answer to either question.

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For a lower bound, observe that the trigonal locus is [unirational][1], hence has Zariski dense rational points, and has dimension $2g+1$. There are many other kinds of obvious rational subvarieties in the moduli space of curves (e.g. parameterizing complete intersections), but they all seem to have lower dimensions.

> For large g, is the trigonal locus the largest such subvariety?

Edit: Felipe pointed out Jason Starr's comment that the trigonal locus is actually larger than the hyperelliptic locus, and has Zariski dense rational points, so I switched hyperelliptic to trigonal in my best guess for the largest subvariety.

  [1]: https://mathoverflow.net/questions/138581/are-most-curves-over-q-pointless/138592#138592