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.

For a lower bound, observe that the trigonal locus is unirational, 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.

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    $\begingroup$ mathoverflow.net/questions/138581/… and in particular Jason Starr's comments. $\endgroup$ – Felipe Voloch Nov 14 '14 at 2:01
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    $\begingroup$ Barry Mazur mentions in this 2011 video [video.mit.edu/watch/problem-session-barry-mazur-8122 ] that the asymptotic fraction $2/3$ realized by the loci of hyperelliptic/trigonal curves appears to be the best known on this type of question. ("If you could improve [this fraction], you would have done something quite siginificant.") $\endgroup$ – Vesselin Dimitrov Nov 14 '14 at 2:29
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    $\begingroup$ @JasonStarr My guess is that the OP has already asked Nick :) $\endgroup$ – Igor Rivin Nov 14 '14 at 13:55
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    $\begingroup$ I believe that one can show that the stack of curves with a map of degree $d$ to a $\mathbb P^1$ is dominated by a rational variety for all $d \le 5$. This would give a lower bound $2g+5$ for the dimension of the closure. $\endgroup$ – Angelo Dec 1 '14 at 18:17
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    $\begingroup$ This follows from the Casnati-Ekedahl description of coverings of degrees 4 and 5 (see G. Casnati, T. Ekedahl: Covers of algebraic varieties I. A general structure theorem, covers of degree 3,4 and Enriques' surfaces. J. Algebraic Geom., 5 (1996), pp. 439-460, and G. Casnati: Covers of algebraic varieties II. Covers of degree 5 and construction of surfaces. J. Algebraic Geom., 5 (1996) pp 461-477) together with the description of the stack of globally generated vector bundles on P^1 in M. Bolognesi, A. Vistoli, Stacks of trigonal curves, Trans. Amer. Math. Soc., 364 (2012), 3365–3393. $\endgroup$ – Angelo Dec 3 '14 at 17:13

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