On the universal degenerating family of Riemann surfaces

• Yukio Matsumoto

Gakushuin University, Tokyo, Japan

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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.