Families of hyperelliptic curves

TL;DR

This paper investigates the structure of hyperelliptic curves' moduli spaces, computing their Picard groups, and exploring the relationships between families and the existence of tautological families.

Contribution

It provides explicit descriptions of the Picard groups of the moduli spaces and analyzes the conditions for families sharing the same modular map and the existence of tautological families.

Findings

Computed Picard groups of the moduli spaces.

Described generators explicitly.

Analyzed conditions for families with identical modular maps.

Abstract

The main subject is the difference between the coarse moduli space and the stack of hyperelliptic curves. In particular, we compute their Picard groups, giving explicit description of the generators. We also study how many families of hyperelliptic curves may have the same modular map as well as the existence of the tautological family over the coarse moduli space of hyperelliptic curves.