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

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 $\mahfrak M_g}$.

27 pages, submitted for the Proceedings volume of "The Conference on Geometry" honoring Shing-Tung Yau's 60th birthday. To be published in the "Advanced Lectures in Mathematics" series of International Press, in cooperation with the Higher Education Press and the Stefan Banach International Mathematical Centre (Institute of Mathematics Polish Academy of Sciences Publishing House)