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A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves. (English) Zbl 0981.58007
The authors determine an expression \(\xi^{s}_g(\gamma)\) for the virtual Euler characteristics of the moduli spaces of \(s\)-pointed real (\(\gamma=1/2\)) and complex (\(\gamma=1\)) algebraic curves. In particular, for the spaces of real curves this gives \((-2)^{s-1}(1-2^{g-1})(g+s-2)!B_g/g!\) for the Euler characteristic, which complements the Harer and Zagier formula \((-1)^{s}(g+s-2)!B_{g+1}/(g+1)(g-1)!\) for the Euler characteristic of the moduli space of complex algebraic curves (\(B_g\) denotes the \(g\)th Bernoulli number). The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to count cells.

MSC:
58D29 Moduli problems for topological structures
58C35 Integration on manifolds; measures on manifolds
30F30 Differentials on Riemann surfaces
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