Combinatorial Classes, Hyperelliptic Loci, and Hodge Integrals

Simple item page
dc.creatorBene, Alex James
dc.date2006-10-19
dc.date.accessioned2026-07-07T07:29:14Z
dc.date.available2026-07-07T07:29:14Z
dc.descriptionA closed formula is obtained for the integral $\int_{\mathcal{\bar{H}}_g^1}κ_{1}ψ^{2g-2}$ 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 $\mathcal{M}^{comb}_{g,1}$, a PL-orbifold whose cells are enumerated by fatgraphs. This cell decomposition can be used to naturally construct combinatorial PL-cycles $W_a\subset\mathcal{M}^{comb}_{g,1}$ whose homology classes are essentially the Poincaré duals of the Mumford-Morita-Miller classes $κ_a$. In this paper we construct another PL-cycle $\mathcal{H}^{comb}_g \subset \mathcal{M}^{comb}_{g,1}$ representing the locus of hyperelliptic Weierstraß points and explicitly describe the chain level intersection of this cycle with $W_1$. Using this description of $\mathcal{H}^{comb}_g\cap W_1$, the duality between Witten cycles $W_a$ and the $κ_a$ classes, and Kontsevich's scheme of integrating $ψ$ classes, the integral $\int_{\mathcal{\bar{H}}_g^1}κ_{1}ψ^{2g-2}$ is reduced to a weighted sum over graphs and is evaluated by the enumeration of trees.
dc.description31 pages, 11 figures
dc.identifierhttps://arxiv.org/abs/math/0610603
dc.identifierhttp://arxiv.org/abs/math/0610603
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/118027
dc.subjectGeometric Topology
dc.subjectMathematical Physics
dc.subjectAlgebraic Geometry
dc.subjectCombinatorics
dc.subject57M99
dc.titleCombinatorial Classes, Hyperelliptic Loci, and Hodge Integrals
dc.typetext

Files

Collections

arXiv