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 Mosc. Math. J., 2007, Volume 7, Number 1, Pages 85–107 (Mi mmj272)

Counting ramified converings and intersection theory on spaces of rational functions. I. Cohomology of Hurwitz spaces

S. Landoa, D. Zvonkineb

a Laboratoire J.-V. Poncelet, Independent University of Moscow
b Institut de Mathématiques de Jussieu

Abstract: The Hurwitz space is a compactification of the space of rational functions of a given degree. The Lyashko–Looijenga map assigns to a rational function the set of its critical values. It is known that the number of ramified coverings of $\mathbb CP^1$ by $\mathbb CP^1$ with prescribed ramification points and ramification types is related to the degree of the Lyashko–Looijenga map on various strata of the Hurwitz space. Here we explain how the degree of the Lyashko–Looijenga map is related to the intersection theory on this space. We describe the cohomology algebra of the Hurwitz space and prove several relations between the homology classes represented by various strata.

Key words and phrases: Riemann surfaces, moduli space, ramified coverings, Lyashko–Looijenga map, Hurwitz space, Hurwitz numbers.

DOI: https://doi.org/10.17323/1609-4514-2007-7-1-85-107

Full text: http://www.ams.org/.../abst7-1-2007.html
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MSC: 05A, 14C, 14D22, 30F
Received: April 12, 2006; in revised form May 27, 2006
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Citation: S. Lando, D. Zvonkine, “Counting ramified converings and intersection theory on spaces of rational functions. I. Cohomology of Hurwitz spaces”, Mosc. Math. J., 7:1 (2007), 85–107

Citation in format AMSBIB
\Bibitem{LanZvo07} \by S.~Lando, D.~Zvonkine \paper Counting ramified converings and intersection theory on spaces of rational functions. I.~Cohomology of Hurwitz spaces \jour Mosc. Math.~J. \yr 2007 \vol 7 \issue 1 \pages 85--107 \mathnet{http://mi.mathnet.ru/mmj272} \crossref{https://doi.org/10.17323/1609-4514-2007-7-1-85-107} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2324558} \zmath{https://zbmath.org/?q=an:1131.14034} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000261708300005} 

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This publication is cited in the following articles:
1. M. E. Kazarian, S. K. Lando, “Thom Polynomials for Maps of Curves with Isolated Singularities”, Proc. Steklov Inst. Math., 258 (2007), 87–99
2. Lando S.K., “Combinatorial facets of Hurwitz numbers”, Applications of group theory to combinatorics, CRC Press, Boca Raton, FL, 2008, 109–131
3. A. Yu. Morozov, “Unitary integrals and related matrix models”, Theoret. and Math. Phys., 162:1 (2010), 1–33
4. Kokotov A., Korotkin D., Zograf P., “Isomonodromic tau function on the space of admissible covers”, Adv. Math., 227:1 (2011), 586–600
5. B. S. Bychkov, “Vychislenie megakart”, Sib. elektron. matem. izv., 10 (2013), 170–179
6. Natanzon S. Zabrodin A., “Symmetric Solutions To Dispersionless 2D Toda Hierarchy, Hurwitz Numbers, and Conformal Dynamics”, Int. Math. Res. Notices, 2015, no. 8, 2082–2110
7. B. S. Bychkov, “Stepeni kogomologicheskikh klassov multiosobennostei v prostranstvakh Gurvitsa ratsionalnykh funktsii”, Funkts. analiz i ego pril., 53:1 (2019), 16–30