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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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##### References:
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