Complete surfaces in $M_g$

Question

Let $M_g$ be the moduli space of genus g curves. In Zaal's paper ("A complete Surface in $M_6$ in Characteristic $> 2$"), the author mentioned that there is a known construction of complete subvarieties of dimension $d\geq 1$ in $M_g$ with $g\geq 2^{d+1}$ but did not give a reference. Is anyone familiar with this construction and/or can point me to a reference?

Answer 1

Theorem 2.33 of Moduli of curves by Harris and Morrison gives a construction with weaker bounds.

The idea is to start with a curve $C$ of genus at least two and consider the family of degree 3 branched coverings ramified at a unique point. This gives a one-dimensional complete family of curves of a certain genus. Iterating this construction, one gets complete families of arbitrary dimension.

Answer 2

If $d:Y\to X$ is an étale double cover, then each point $p$ of $X$ gives a pair $\{a_p,b_p\}=d^{-1}(p)$ of distinct points in $Y$.

Given such a $\{a_p,b_p\}$ there are finitely many double covers $e:Z_p\to Y$ which are branched at precisely these two points. The number of covers does not depend on the choice of pairs.

If $X$ has genus $g$, then $Y$ has genus $2g-1$ and $Z_p$ has genus $h=2(2g-1)$.

This gives a family $\tilde{Z}\to\tilde{X}$ of curves $Z_p$ parametrised by an étale cover $\tilde{X}$ of $X$. (The cover $\tilde{X}$ is required to take care of the finitely many choices.) One needs to show that (generically) if $(X,p)$ moves in the moduli space $M_{g,1}$ with dimension $k$ then the cover $Z_p$ moves in the moduli $M_h$ with the same dimension $k$. Note that we need to start with $g\geq 2$ for there to be a positive dimensional complete family of $(X,p)$.

Now, this construction can be iterated replacing $X$ by $Z_p$ which is the fibre of $\tilde{Z}\times_{\tilde{X}}\tilde{Z}$ over $\tilde{Z}$. The point $p$ is replaced by the "diagonal" section $\delta$ so that the pair $(Z_p,\delta)$ now varies over a $2$-dimensional base.

This gives a surface parametrising curves of genus $2(2\cdot 2(2g-1)-1)$.

One can iterate this construction to get familes of dimension $d$ that parametrise curves of genus $g$ which grows like $2^{2d}$.

This is worse than the growth that the original question posed, but perhaps someone can improve on it!