# On the universal degenerating family of Riemann surfaces

### Yukio Matsumoto

Gakushuin University, Tokyo, Japan

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## Abstract

Let $Σ_{g}$ be a closed oriented (topological) surface of genus $g$ ($≧2$). Over the Teichmüller space $T(Σ_{g})$ of $Σ_{g}$, Bers constructed a universal family $V(Σ_{g})$ of curves of genus $g$, which would be well called “the tautological family of Riemann surfaces”. The mapping class group $Γ_{g}$ of $Σ_{g}$ acts on $V(Σ_{g})→T(Σ_{g})$ in a fibration preserving manner. Dividing the fiber space by this action, we obtain an “orbifold fiber space” $Y(Σ_{g})→M(Σ_{g})$, where $Y(Σ_{g})$ and $M(Σ_{g})$ denote $V(Σ_{g})/Γ_{g}$ and $T(Σ_{g})/Γ_{g}$, respectively. The latter quotient $M(Σ_{g})$ is called the *moduli space* of $Σ_{g}$. The fiber space $Y(Σ_{g})→M(Σ_{g})$ can be naturally compactified to another orbifold fiber space $Y(Σ_{g}) →M(Σ_{g}) $. The base space $M(Σ_{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 $M(Σ_{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 $Y(Σ_{g}) →M(Σ_{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.