Irreducibility of the space of cyclic covers of algebraic curves of fixed numerical type and the irreducible components of $Sing (\bar{\mathfrak M_g})$

TL;DR

This paper proves the irreducibility of spaces of cyclic covers of algebraic curves with fixed numerical types and describes the irreducible components of the singular locus of the compactified moduli space of curves, extending prior work.

Contribution

It establishes irreducibility results for cyclic covers of algebraic curves and characterizes the singular locus components of the moduli space of curves.

Findings

Proved irreducibility of cyclic cover spaces with fixed numerical type.

Described irreducible components of the singular locus of ar{\u2113}mathfrak{M}_g.

Extended Cornalba's work on the singular locus of moduli spaces.

Abstract

We prove irreducibility for the space of cyclic covers of fixed numerical type between smooth projective curves, and also for the space of cyclic covers of prime order and of fixed numerical-combinatorial type between moduli-stable projective curves. As an application we describe the irreducible components of the singular locus of the compactified moduli space of curves $\bar{\mahfrak M_g}$, extending the work of Cornalba, who described the irreducible components of the singular locus of the moduli space of curves $\mahfrak M_g}$.