Complete moduli for fibered surfaces

Abstract

Families of stable curves of genus $γ$ over a smooth curve $C$ correspond to morphisms from $C$ to the moduli stack of stable curves $\bar{\cal M}_γ$. It is natural to compactify the corresponding moduli problem using stable maps into the stack. In order to get a complete moduli problem, the source curves must acquire extra structure, and you will have to read the paper to find what that is. In this paper, we define these stable maps into $\bar{\cal M}_γ$ in characteristic 0, and show that they form a proper Deligne-Mumford stack admitting a projective coarse moduli space. A comparison with Alexeev's work is given. Natural generalizations for stable maps into other Deligne-Mumford stacks will appear in a follow up paper, along with applications to areas such as admissible covers, level structures and higher dimensional families.