On the universal degenerating family of Riemann surfaces

Abstract

Let ${\Sigma }_g$ be a closed oriented (topological) surface of genus $g$ ($\geqq 2$). Over the Teichmüller space $T({\Sigma }_g)$ of ${\Sigma }_g$, Bers constructed a universal family $V({\Sigma }_g)$ of curves of genus $g$, which would be well called “the tautological family of Riemann surfaces”. The mapping class group ${\Gamma }_g$ of ${\Sigma }_g$ acts on $V({\Sigma }_g)\to T({\Sigma }_g)$ in a fibration preserving manner. Dividing the fiber space by this action, we obtain an “orbifold fiber space” $Y({\Sigma }_g)\to M({\Sigma }_g)$, where $Y({\Sigma }_g)$ and $M({\Sigma }_g)$ denote $V({\Sigma }_g)/{\Gamma }_g$ and $T({\Sigma }_g)/{\Gamma }_g$, respectively. The latter quotient $M({\Sigma }_g)$ is called the moduli space of ${\Sigma }_g$. The fiber space $Y({\Sigma }_g)\to M({\Sigma }_g)$ can be naturally compactified to another orbifold fiber space $\overline{Y({\Sigma }_g)}\to \overline{M({\Sigma }_g)}$. The base space $\overline{M({\Sigma }_g)}$ is called the Deligne–Mumford compactification. Since this compactification is constructed by adding “stable curves” at infinity, it is usually accepted that the compactified moduli space $\overline{M({\Sigma }_g)}$ is the coarse moduli space of stable curves of genus $g$. In this paper, we will sketch our argument which leads to a conclusion, somewhat contradictory to the above general acceptance, that the compactified family $\overline{Y({\Sigma }_g)}\to \overline{M({\Sigma }_g)}$ is the universal degenerating family of Riemann surfaces, i.e., it virtually parametrizes not only stable curves but also all types of degenerate and non-degenerate curves.

Our argument is a combination of the Bers–Kra theory and Ashikaga’s precise stable reduction theorem.