# The locus of cyclic covers in the moduli space of curves

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 $W\subset Z$.

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