A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves

Abstract

We determine an expression 9 (1X) for the virtual Euler characteristics of the moduli spaces of s-pointed real (-y = 1/2) and complex (-y = 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(1-29-1)(g?s-2)!Bg/g! where Bg 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+ /(g+1) (g -1) ! The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to counit cells. The approach involves a parameter y that permits specialization of the formula to the real and complex cases. This suggests that 4 (-y) itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.