What I mean by the (slightly facetious) title is:

The classical theory of algebraic curves from the 19th century was split in two in the 20th century (much like the theory of groups): the theory of abstract curves and the theory of embedded curves.

In the case of $\mathcal{M}_g$, there is a well developed method (the Brill-Noether theory) for realizing the universal curve over the moduli space (except some loci of real codimension $\geq 2$ which are not important) as a family of curves embedded in some (possibly non-trivial) projective space bundle over $\mathcal{M}_g$. This family of embedded curves would have monodromy group $Mod_g$.

What happens when I consider the level k in the Johnson filtration $Mod_g[k]$? Are there more interesting ways to generate a family of embedded curves $\mathcal{X}/\mathcal{B}$ (+ perhaps some decorations like homology or nilpotent markings?) with $\mathcal{B}$ algebraic (or analytic) such that the map which sends them into the k-th Torelli space (in the sense of Morita-Penner: the quotient of Teichmuller space by the action of $Mod_g[k]$) is surjective?