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


Authors: I. P. Goulden, J. L. Harer and D. M. Jackson
Journal: Trans. Amer. Math. Soc. 353 (2001), 4405-4427
MSC (2000): Primary 58D29, 58C35; Secondary 05C30, 05E05
DOI: https://doi.org/10.1090/S0002-9947-01-02876-8
Published electronically: July 9, 2001
MathSciNet review: 1851176
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Abstract:

We 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 space of real curves of genus $g$ with a fixed point free involution, we find that the Euler characteristic is $(-2)^{s-1}(1-2^{g-1})(g+s-2)!B_g/g!$ where $B_g$ is the $g$th Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is $(-1)^{s}(g+s-2)!B_{g+1}/(g+1)(g-1)!$

The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to count cells. The approach involves a parameter $\gamma$ that permits specialization of the formula to the real and complex cases. This suggests that $\xi^s_g(\gamma)$ itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.


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Additional Information

I. P. Goulden
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
Email: ipgoulden@math.uwaterloo.ca

J. L. Harer
Affiliation: Department of Mathematics, Duke University, Durham, North Carolina 27708-0320
Email: harer@math.duke.edu

D. M. Jackson
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
Email: dmjackson@math.uwaterloo.ca

DOI: https://doi.org/10.1090/S0002-9947-01-02876-8
Received by editor(s): January 22, 1999
Received by editor(s) in revised form: April 7, 1999
Published electronically: July 9, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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