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The Weil-Petersson volume of the moduli space of punctured spheres. (English) Zbl 0792.32016
Bödigheimer, Carl-Friedrich (ed.) et al., Mapping class groups and moduli spaces of Riemann surfaces. Proceedings of workshops held June 24-28, 1991, in Göttingen, Germany, and August 6-10, 1991, in Seattle, WA (USA). Providence, RI: American Mathematical Society. Contemp. Math. 150, 367-372 (1993).
Let $${\mathcal M}_{0,n}$$ be the moduli space of Riemann surfaces of genus 0 with $$n \geq 3$$ ordered punctures. It is a well-known fact that the Weil- Petersson volume of $${\mathcal M}_{0,n}$$ is finite [H. Masur, Duke Math. J. 43, 623-635 (1976; Zbl 0358.32017)].
The aim of this paper is to prove the following result $\text{Vol}_{WP} ({\mathcal M}_{0,n})={\pi^{2(n-3)} \over (n-3)!}V_ n,\;n \geq 3,$ where the integral numbers $$V_ n$$ for $$n \geq 4$$ are given by the recursion relation \begin{aligned} V_ n & = {1 \over 2} \sum^{n- 3}_{i=1} {i(n-i-2) \over n-1} {n-4 \choose i-1} {n \choose i+1} V_{i+2} V_{n-i}, \\ V_ 3 & = 1.\end{aligned} In particular $$V_ 4=1$$, $$V_ 5=5$$, $$V_ 6=61$$, $$V_ 7=1379$$, $$V_ 8=49946,\dots$$.
For the entire collection see [Zbl 0777.00025].

##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 14H10 Families, moduli of curves (algebraic)