A geometric parametrization for the virtual euler characteristics of the moduli spaces of real and complex algebraic curves
We determine an expression ξgs(γ) for the virtual Euler characteristics of the moduli spaces of s-pointed real (7 = 1/2) and complex (7 = 1) algebraic curves. In particular, for the space of real curves of genus g with a fixed point free involution, we find that the Euler characteristic is (-2)s-1(1-2q-1)(g+s-2)!Bg/g! where gth is the gth Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is (-1)s(g+s-2)!Bg+1/(g+1)(g- 1)! The proof uses Strcbel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to count cells. The approach involves a parameter γ that permits specialization of the formula to the real and complex cases. This suggests that ξgs(γ) itself may describe the Eulcr characteristics of some related moduli spaces, although we do not yet know what these spaces might be. ©2001 American Mathematical Society.
Goulden, IP; Harer, JL; Jackson, DM
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