On the automorphisms of moduli spaces of curves

TL;DR

This paper surveys recent results on automorphisms of moduli spaces of curves, extending techniques to intermediate and degenerate cases, providing a comprehensive overview of the automorphism groups in these geometric contexts.

Contribution

It extends previous work on automorphisms of moduli spaces to include intermediate and degenerate cases, broadening understanding of their symmetries.

Findings

Automorphism groups of various moduli spaces are characterized.

Techniques are extended to spaces with zero weights.

Results unify understanding of automorphisms across different moduli space compactifications.

Abstract

In the last years the biregular automorphisms of the Deligne-Mumford's and Hassett's compactifications of the moduli space of n-pointed genus g smooth curves have been extensively studied by A. Bruno and the authors. In this paper we give a survey of these recent results and extend our techniques to some moduli spaces appearing as intermediate steps of the Kapranov's and Keel's realizations of $\bar{M}_{0,n}$, and to the degenerations of Hassett's spaces obtained by allowing zero weights.