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Then one may wonder if something could be said for higher dimensional bases. The direct generalization of the original question would be to replace an elliptic curve with an abelian variety. The same statement holds, (in fact a little more than that is) proved in Families over a base with a birationally nef tangent bundle Mathematische Annalen, 1997, Volume 308, Number 2, Pages 347-359

Viehweg's conjecture is currently known for proper base varieties by Families of varieties of general type over compact bases, Advances in Mathematics, Volume 218, Issue 3, 20 June 2008, Pages 649–652 and Viehwegʼs hyperbolicity conjecture is true over compact bases, Advances in Mathematics, Volume 229, Issue 3, 15 February 2012, Pages 1640–1642 and over (up to) $3$-dimensional bases in general by The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties, Duke Math. J. Volume 155, Number 1 (2010), 1-33..

Then one may wonder if something could be said for higher dimensional bases. The direct generalization of the original question would be to replace an elliptic curve with an abelian variety. The same statement holds, (in fact a little more than that is) proved in Families over a base with a birationally nef tangent bundle Mathematische Annalen, 1997, Volume 308, Number 2, Pages 347-359

Viehweg's conjecture is currently known for proper base varieties by Families of varieties of general type over compact bases, Advances in Mathematics, Volume 218, Issue 3, 20 June 2008, Pages 649–652 and Viehwegʼs hyperbolicity conjecture is true over compact bases, Advances in Mathematics, Volume 229, Issue 3, 15 February 2012, Pages 1640–1642 and over (up to) $3$-dimensional bases in general by The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties, Duke Math. J. Volume 155, Number 1 (2010), 1-33..

Here "log general type" means that if $X$ is the variety in question and $\overline X$ is a projective variety such that $X\subseteq \overline X$ is an open subset and $D=\overline X\setminus X$ is a divisor, then $\omega_{\overline X}(D)$ is big (=has maximal Kodaira dimension).

Here log general type means that if $X$ is the variety in question and $\overline X$ is a projective variety such that $X\subseteq \overline X$ is an open subset and $D=\overline X\setminus X$ is a divisor, then $\omega_{\overline X}(D)$ is big (=has maximal Kodaira dimension).

As far as the coarse space goes, it is probably more of a curiosity, but it still seems interesting. Oort proved that the coarse space of $M_g$ actually contains rational curves. And then there are many results concerning the question of what kind of complete subvarieties might $M_g$ have. Or what about complete subvarieties through a general point? There are various results in this direction, but I am already digressing....

As far as the coarse space goes, it is probably more of a curiosity, but it still seems interesting. Oort proved that the coarse space of $M_g$ actually contains rational curves. Perhaps his proof can be adapted to prove the same for an elliptic curve. (The main idea is to construct a family over a curve with a given map to $\mathbb P^1$ such that fibers over the points mapping to the same point on $\mathbb P^1$ are isomorphic, so the moduli map factors through the map to $\mathbb P^1$).

And then there are many results concerning the question of what kind of complete subvarieties might $M_g$ have. Or what about complete subvarieties through a general point? There are various results in this direction, but I am already digressing....