Genus and Hurwitz numbers: Recursions, formulas, and graph-theoretic interpretations

Author:
Ravi Vakil

Journal:
Trans. Amer. Math. Soc. **353** (2001), 4025-4038

MSC (2000):
Primary 14H10, 05C30; Secondary 58D29

Published electronically:
June 1, 2001

MathSciNet review:
1837218

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus and . (Hurwitz numbers essentially count irreducible genus covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden, Jackson and Vainshtein, and extend results of Hurwitz and many others.

**[A]**V. I. Arnol′d,*Topological classification of complex trigonometric polynomials and the combinatorics of graphs with an identical number of vertices and edges*, Funktsional. Anal. i Prilozhen.**30**(1996), no. 1, 1–17, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl.**30**(1996), no. 1, 1–14. MR**1387484**, 10.1007/BF02383392**[CT]**Michael Crescimanno and Washington Taylor,*Large 𝑁 phases of chiral 𝑄𝐶𝐷₂*, Nuclear Phys. B**437**(1995), no. 1, 3–24. MR**1354345**, 10.1016/0550-3213(94)00561-R**[D]**József Dénes,*The representation of a permutation as the product of a minimal number of transpositions, and its connection with the theory of graphs*, Magyar Tud. Akad. Mat. Kutató Int. Közl.**4**(1959), 63–71 (English, with Hungarian and Russian summaries). MR**0115936****[ELSV1]**Torsten Ekedahl, Sergei Lando, Michael Shapiro, and Alek Vainshtein,*On Hurwitz numbers and Hodge integrals*, C. R. Acad. Sci. Paris Sér. I Math.**328**(1999), no. 12, 1175–1180 (English, with English and French summaries). MR**1701381**, 10.1016/S0764-4442(99)80435-2**[ELSV2]**T. Ekedahl, S. Lando, M. Shapiro and A. Vainshtein,*Hurwitz numbers and intersections on moduli spaces of curves*, preprint 2000, math. AG/0004096.**[EK]**L. Ernström and G. Kennedy,*Contact cohomology of the projective plane*, Amer. J. Math.**121**(1999), no. 1, 73-96. CMP**99:16****[FP]**W. Fulton and R. Pandharipande,*Notes on stable maps and quantum cohomology*, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96. MR**1492534**, 10.1090/pspum/062.2/1492534**[GL]**V. V. Goryunov and S. K. Lando,*On enumeration of meromorphic functions of the line*, The Arnoldfest (Toronto, 1997), Fields Inst. Commun.**24**, A.M.S., Providence, R.I., 1999. CMP**2000:08****[GJ1]**I. P. Goulden and D. M. Jackson,*Transitive factorisations into transpositions and holomorphic mappings on the sphere*, Proc. Amer. Math. Soc.**125**(1997), no. 1, 51–60. MR**1396978**, 10.1090/S0002-9939-97-03880-X**[GJ2]**I. P. Goulden and D. M. Jackson,*A proof of a conjecture for the number of ramified coverings of the sphere by the torus*, J. Combin. Theory Ser. A**88**(1999), no. 2, 246–258. MR**1723796**, 10.1006/jcta.1999.2992**[GJ3]**I. P. Goulden and D. M. Jackson,*The number of ramified coverings of the sphere by the double torus, and a general form for higher genera*, J. Combin. Theory Ser. A**88**(1999), no. 2, 259–275. MR**1723797**, 10.1006/jcta.1999.2993**[GJVn]**I. P. Goulden, D. M. Jackson and A. Vainshtein,*The number of ramified coverings of the sphere by the torus and surfaces of higher genera*, Annals of Comb.**4**(2000), 27-46. CMP**2000:14****[GJV]**I. P. Goulden, D. M. Jackson and R. Vakil,*The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals*, Proc. London Math. Soc., to appear.**[GP]**T. Graber and R. Pandharipande, personal communication.**[GV]**T. Graber and R. Vakil,*Hodge integrals, Hurwitz numbers, and virtual localization*, preprint 2000, math. AG/0003028, submitted for publication.**[H]**A. Hurwitz,*Über die Anzahl der Riemann'schen Flächen mit gegebenen Verzweigungspunkten*, Math. Ann.**55**(1902) 53-66.**[L]**László Lovász,*Combinatorial problems and exercises*, 2nd ed., North-Holland Publishing Co., Amsterdam, 1993. MR**1265492****[P]**Rahul Pandharipande,*Intersections of 𝐐-divisors on Kontsevich’s moduli space \overline𝐌_{0,𝐧}(𝐏^{𝐫},𝐝) and enumerative geometry*, Trans. Amer. Math. Soc.**351**(1999), no. 4, 1481–1505. MR**1407707**, 10.1090/S0002-9947-99-01909-1**[Sch]**Michael Schlessinger,*Functors of Artin rings*, Trans. Amer. Math. Soc.**130**(1968), 208–222. MR**0217093**, 10.1090/S0002-9947-1968-0217093-3**[SSV]**B. Shapiro, M. Shapiro, and A. Vainshtein,*Ramified coverings of with one degenerate branching point and enumeration of edge-ordered graphs*, Topics in singularity theory, 219-227, Amer. Math. Soc. Transl. Ser. 2, 180, Amer. Math. Soc., Providence, RI, 1997. CMP**2000:15****[St]**V. Strehl,*Minimal transitive products of transpositions -- the reconstruction of a proof of A. Hurwitz*, Sém. Lothar. Combin.**37**(1996), Art. S37c, 12 pp. (electronic, see http://cartan.u-strasbg.fr:80/ slc/).**[V1]**R. Vakil,*Recursions for characteristic numbers of genus one plane curves*, Arkiv för Matematik, to appear.**[V2]**R. Vakil,*The enumerative geometry of rational and elliptic curves in projective space*, J. Reine Angew. Math.,**529**(2000), 101-153. CMP**2001:05**

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Additional Information

**Ravi Vakil**

Affiliation:
Department of Mathematics, Stanford University, Building 380, MC2125, Stanford, California 94305

Email:
vakil@math.stanford.edu

DOI:
https://doi.org/10.1090/S0002-9947-01-02776-3

Received by editor(s):
December 16, 1998

Published electronically:
June 1, 2001

Additional Notes:
The author was supported in part by NSF Grant DMS-9970101

Article copyright:
© Copyright 2001
American Mathematical Society