Lectures on the ELSV formula

Details Lectures on the ELSV formula Lectures on the ELSV formula

2010-04-06

arxiv.org/abs/1004.0853v1

Transformation Groups and Moduli Spaces of Curves, 195--216, Adv. Lect. Math. (ALM) 16, Higher Education Press and Internation

Mathematics

Algebraic Geometry

25 pages

Scientific paper

The ELSV formula, first proved by Ekedahl, Lando, Shapiro, and Vainshtein, relates Hurwitz numbers to Hodge integrals. Graber and Vakil gave another proof of the ELSV formula by virtual localization on moduli spaces of stable maps to the projective line, and also explained how to simplify their proof using moduli spaces of relative stable maps to the projective line relative to a point. In this expository article, we explain what the ELSV formula is and how to prove it by virtual localization on moduli spaces of relative stable maps, following Graber-Vakil. This note is based on lectures given by the author at Summer School on "Geometry of Teichmuller Spaces and Moduli Spaces of Curves" at Center of Mathematical Sciences, Zhejiang University, July 14--20, 2008.

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