Combinatorial Classes, Hyperelliptic Loci, and Hodge Integrals

Abstract

A closed formula is obtained for the integral ${\int }_{{\bar{\mathcal{H}}}_g^1}{\kappa }_1{\psi }^{2g-2}$ of tautological classes over the locus of hyperelliptic Weierstraß points in the moduli space of curves. As a corollary, a relation between Hodge integrals is obtained. The calculation utilizes the homeomorphism between the moduli space of curves ${\mathcal{M}}_{g,1}$ and the combinatorial moduli space ${\mathcal{M}}_{g,1}^{comb}$, a PL-orbifold whose cells are enumerated by fatgraphs. This cell decomposition can be used to naturally construct combinatorial PL-cycles $W_a\subset {\mathcal{M}}_{g,1}^{comb}$ whose homology classes are essentially the Poincaré duals of the Mumford-Morita-Miller classes ${\kappa }_a$. In this paper we construct another PL-cycle ${\mathcal{H}}_g^{comb}\subset {\mathcal{M}}_{g,1}^{comb}$ representing the locus of hyperelliptic Weierstraß points and explicitly describe the chain level intersection of this cycle with $W_1$. Using this description of ${\mathcal{H}}_g^{comb}\cap W_1$, the duality between Witten cycles $W_a$ and the ${\kappa }_a$ classes, and Kontsevich's scheme of integrating $\psi$ classes, the integral ${\int }_{{\bar{\mathcal{H}}}_g^1}{\kappa }_1{\psi }^{2g-2}$ is reduced to a weighted sum over graphs and is evaluated by the enumeration of trees.

31 pages, 11 figures

References in corpus (1)

• Strebel differentials on stable curves and Kontsevich's proof of Witten's conjecture