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 $W\subset Z$.