Question

Let $\mathcal{M}_g$ be the moduli space of smooth curves of genus $g$. Let $Z$ be the closure in $\mathcal{M}_g$ of the set of smooth curves of genus $g$ which are a cyclic cover of the projective line.

Question. Is $Z$ irreducible?

Question. What is the dimension of $Z$? Do we have non-trivial bounds?

Question. Is $Z$ affine?

Remark. Let $W$ be the closure of the set of smooth curves which are a cyclic cover of the projective line of prime degree. Then it is known that $W$ is affine. Note that $W\subset Z$.

Answer 1

The following paper:

M. Cornalba, On the locus of curves with automorphisms, Annali di Matematica pura ed applicata (4) 149 (1987), 135-151. Erratum, Annali di Matematica pura ed applicata (4) 187 (2008), 185-186. (A revised version incorporating the changes described in the Erratum is available on the author's web page)

contains an explicit description of the irreducible components of the locus of curves with an automorphism, including computation of the dimensions.