Genus and Hurwitz numbers: Recursions, formulas, and graphtheoretic interpretations
Author:
Ravi Vakil
Journal:
Trans. Amer. Math. Soc. 353 (2001), 40254038
MSC (2000):
Primary 14H10, 05C30; Secondary 58D29
Published electronically:
June 1, 2001
MathSciNet review:
1837218
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: We derive a closedform expression for all genus 1 Hurwitz numbers, and give a simple new graphtheoretic 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.
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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.ustrasbg.fr:80/ slc/).
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R. Vakil, Recursions for characteristic numbers of genus one plane curves, Arkiv för Matematik, to appear.
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R. Vakil, The enumerative geometry of rational and elliptic curves in projective space, J. Reine Angew. Math., 529 (2000), 101153. CMP 2001:05
 [A]
 V. I. Arnol'd, Topological classification of trigonometric polynomials and combinatorics of graphs with an equal number of vertices and edges, Functional analysis and its applications 30 no. 1 (1996) 117. MR 97d:32053
 [CT]
 M. Crescimanno and W. Taylor, Large phases of chiral QCD, Nuclear Phys. B 437 no. 1 (1995) 324. MR 96g:81223
 [D]
 J. Dénes, The representation of a permutation as the product of a minimal number of transpositions and its connection with the theory of graphs, Publ. Math. Inst. Hungar. Acad. Sci. 4 (1959) 6370. MR 22:6733
 [ELSV1]
 T. Ekedahl, S. Lando, M. Shapiro and A. Vainshtein, On Hurwitz numbers and Hodge integrals, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 12, 11751180.MR 2001b:14083
 [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, 7396. CMP 99:16
 [FP]
 W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in Algebraic geometry Santa Cruz 1995 v. 2, J. Kollár, R. Lazarsfeld, D. Morrison eds., A.M.S., Providence, 1997. MR 98m:14025
 [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. of the A.M.S. 125 no. 1 (1997) 5160. MR 97j:05007
 [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, 246258. MR 2000i:05009
 [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), 259275. MR 2000i:05010
 [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), 2746. CMP 2000:14
 [GJV]
 I. P. Goulden, D. M. Jackson and R. Vakil, The GromovWitten 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) 5366.
 [L]
 L. Lovasz, Combinatorial Problems and Exercises 2nd ed., Akadémiai Kiadó, Budapest, 1993. MR 94m:05001
 [P]
 R. Pandharipande, Intersection of divisors on Kontsevich's moduli space and enumerative geometry, Trans. A.M.S., 351 (1999), no. 4, 14811505. MR 99f:14068
 [Sch]
 M. Schlessinger, Functors of Artin rings, Trans. A.M.S. 130 (1968), 208222. MR 36:184
 [SSV]
 B. Shapiro, M. Shapiro, and A. Vainshtein, Ramified coverings of with one degenerate branching point and enumeration of edgeordered graphs, Topics in singularity theory, 219227, 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.ustrasbg.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), 101153. 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:
http://dx.doi.org/10.1090/S0002994701027763
PII:
S 00029947(01)027763
Received by editor(s):
December 16, 1998
Published electronically:
June 1, 2001
Additional Notes:
The author was supported in part by NSF Grant DMS9970101
Article copyright:
© Copyright 2001
American Mathematical Society
