On the universal degenerating family of Riemann surfaces

  • Yukio Matsumoto

    Gakushuin University, Tokyo, Japan
On the universal degenerating family  of Riemann surfaces cover

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Let Σg\Sigma_g be a closed oriented (topological) surface of genus gg (2\geqq 2). Over the Teichmüller space T(Σg)T(\Sigma_g) of Σg\Sigma_g, Bers constructed a universal family V(Σg)V(\Sigma_g) of curves of genus gg, which would be well called “the tautological family of Riemann surfaces”. The mapping class group Γg\Gamma_g of Σg\Sigma_g acts on V(Σg)T(Σg)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(Σg)M(Σg)Y(\Sigma_g) \to M(\Sigma_g), where Y(Σg)Y(\Sigma_g) and M(Σg)M(\Sigma_g) denote V(Σg)/ΓgV(\Sigma_g)/\Gamma_g and T(Σg)/ΓgT(\Sigma_g)/\Gamma_g, respectively. The latter quotient M(Σg)M(\Sigma_g) is called the moduli space of Σg\Sigma_g. The fiber space Y(Σg)M(Σg)Y(\Sigma_g) \to M(\Sigma_g) can be naturally compactified to another orbifold fiber space Y(Σg)M(Σg)\overline{Y(\Sigma_g)} \to \overline{M(\Sigma_g)}. The base space M(Σg)\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 M(Σg)\overline{M(\Sigma_g)} is the coarse moduli space of stable curves of genus gg. 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(Σg)M(Σg)\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.