Hurwitz numbers and intersections on moduli spaces of curves

By Torsten Ekedahl, Sergei K. Lando, Michael Shapiro, A. Vainshtein. 2000

389 citations recorded by the source corpus.

This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/edee0271c991f929e874ae1084099ffbaab9cbd2

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In Geometry and Topology

Citations in the corpus

The Equivariant Gromov-Witten theory of P**1 (2002, methodology): “The operator formalism for Hodge integrals relies crucially upon a formula due to Ekedahl, Lando, Shapiro, and Vainstein (see [6] [7] [11] and also [21]) expressing basic Hurwitz numbers as Hodge integrals.”
K3 surfaces with Picard number one and infinitely many rational points (2005, methodology): “This result, the most technical in this paper, is established by computing the genus 0, one-pointed invariants via localization, together with the use of some beautiful Hodge integral computations from [Faber and Pandharipande 2000; Ekedahl et al. 2001; Graber and Vakil 2003a].”
Equivalence of ELSV and Bouchard–Mariño conjectures for $$r$$r-spin Hurwitz numbers (2013, methodology): “The first is a generalization of the celebrated ELSV formula relating Hurwitz numbers and Hodge integrals [15] that we call the r-ELSV conjecture.”
Polynomiality of monotone Hurwitz numbers in higher genera (2012, methodology): “Hurwitz numbers is a solution of the KP hierarchy, and as shown by Kazarian and Lando [12,13] via the ELSV formula [1], this further implies the celebrated Witten–Kontsevich theorem [14,22] relating intersection theory on moduli spaces to integrable systems.”
Polynomial recursion formula for linear Hodge integrals (2009, neutral): “In terms of the representation we have ρ(γr) = (ab) ∈ Sd.”
Hodge Integrals and Hurwitz Numbers via Virtual Localization (2000, background): “Ekedahl, Lando, Shapiro and Vainshtein have proved a remarkable formula ([ELSV1, Theorem 1.1], [ ELSV2 , Theorem 1.1]; Theorem 2.2 below) linking Hurwitz numbers to Hodge integrals in a particularly elegant way.”
Abelian Hurwitz-Hodge integrals (2008, methodology): “The ELSV formula [9] for linear Hodge integrals on the moduli space of curves arises from the a = 1 specialization of Theorem 1, Hg(μ) = (2g − 2 + d + #)! |Aut(μ)| # ∏”
An algebro-geometric proof of Witten's conjecture (2006, background): “Our proof is based on the following, now well-known, properties of these numbers: • the ELSV formula [4, 5], see also [8], [13], relating Hurwitz numbers to the intersection theory on moduli spaces; • the relationship between Hurwitz numbers and integrable hierarchies conjectured by Pandharipande [1”
Classical Hurwitz numbers and related combinatorics (2016, methodology): “Another essential statement of Theorem 3, namely the fact that H g is a polynomial in 1/(1 − T ) whenever 2g + n − 2 > 0, and that Hg for g ≥ 2 has the form (34), are not new results, but were already proved by Goulden, Jackson and Vakil [28] using the celebrated Ekedahl–Lando–Shapiro–Vainshtein (EL”