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 CP1 by CP1 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.
Received:April 12, 2006; in revised form May 27, 2006
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
This publication is cited in the following 11 articles:
Theo Douvropoulos, “Lyashko–Looijenga morphisms and primitive factorizations of Coxeter elements”, Math. Ann., 2024
Kazarian M., Lando S., Zvonkine D., “Double Hurwitz Numbers and Multisingularity Loci in Genus 0”, Int. Math. Res. Notices, 2022:12 (2022), 9529–9570
Andrea Brini, Karoline van Gemst, “Mirror symmetry for extended affine Weyl groups”, Journal de l'École polytechnique — Mathématiques, 9 (2022), 907
B. S. Bychkov, “Stepeni kogomologicheskikh klassov multiosobennostei v prostranstvakh Gurvitsa ratsionalnykh funktsii”, Funkts. analiz i ego pril., 53:1 (2019), 16–30
B. S. Bychkov, “Degrees of Cohomology Classes of Multisingularities in Hurwitz Spaces of Rational Functions”, Funct Anal Its Appl, 53:1 (2019), 11
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
B. S. Bychkov, “Vychislenie megakart”, Sib. elektron. matem. izv., 10 (2013), 170–179
Kokotov A., Korotkin D., Zograf P., “Isomonodromic tau function on the space of admissible covers”, Adv. Math., 227:1 (2011), 586–600
A. Yu. Morozov, “Unitary integrals and related matrix models”, Theoret. and Math. Phys., 162:1 (2010), 1–33
Lando S.K., “Combinatorial facets of Hurwitz numbers”, Applications of group theory to combinatorics, CRC Press, Boca Raton, FL, 2008, 109–131
M. E. Kazarian, S. K. Lando, “Thom Polynomials for Maps of Curves with Isolated Singularities”, Proc. Steklov Inst. Math., 258 (2007), 87–99
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