Champs de Hurwitz
[Hurwitz stacks]
Mémoires de la Société Mathématique de France, no. 125-126 (2011) , 219 p.

In this work, we give a thorough study of Hurwitz stacks and associated Hurwitz moduli spaces, both in the Galois and the non Galois case, with particular attention to correspondances between these moduli spaces. We compare our construction to those proposed by Abramovich-Corti-Vistoli, Harris-Mumford, and Mochizuki-Wewers. We apply our results to revisit some classical examples, particularly the stacks of stable curves equipped with an arbitrary level structure, and the stacks of tamely ramified cyclic covers. In a second part we exhibit some tautological bundles and cohomology classes naturally living on Hurwitz stacks, and give some universal relations, in particular a higher analogue of the Riemann-Hurwitz formula, between these classes. Applications are given to the stack of cyclic covers of the projective line, with special attention to Cornalba-Harris type relations and to cyclic, in particular hyperelliptic, Hodge integrals.

Dans ce travail, nous effectuons une étude détaillée des champs de Hurwitz et de leurs espaces de modules, tant dans le cas galoisien que dans le cas non galoisien, avec une attention particulière portée aux correspondances entre ces espaces de modules. Nous comparons notre construction à celles proposées par Abramovich-Corti-Vistoli, Harris-Mumford, et Mochizuki-Wewers. Nous mettons en application nos résultats pour revisiter des exemples classiques, notamment les champs de courbes stables munies d’une structure de niveau arbitraire, et les champs de revêtements cycliques modérément ramifiés. Dans une deuxième partie, nous mettons en évidence des fibrés tautologiques et des classes de cohomologie qui vivent naturellement sur les champs de Hurwitz, et nous donnons des relations universelles, dont un analogue supérieur de la formule de Riemann-Hurwitz, entre ces classes. Nous donnons des applications au champ des revêtements cycliques de la droite projective, avec un intérêt particulier pour des relations du type de la relation de Cornalba-Harris et pour les intégrales de Hodge cycliques, notamment hyperelliptiques.

DOI: 10.24033/msmf.437
Classification: 14H10, 14H30, 14H37, 14A20, 14C17, 11G20, 11G30, 14C40
Mot clés : Courbe algébrique, revêtement, revêtement galoisien, champ de Hurwitz, compactification, structure de niveau, Riemann-Hurwitz, revêtement cyclique, classes tautologiques
Keywords: Algebraic curve, covering, Galois covering, algebraic stack, compactification, level structure, Riemann-Hurwitz, cyclic covering, tautological classes
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     author = {Bertin, Jos\'e and Romagny, Matthieu},
     title = {Champs de {Hurwitz}},
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     publisher = {Soci\'et\'e math\'ematique de France},
     number = {125-126},
     year = {2011},
     doi = {10.24033/msmf.437},
     mrnumber = {2920693},
     zbl = {1242.14025},
     language = {fr},
     url = {http://archive.numdam.org/item/MSMF_2011_2_125-126__1_0/}
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Bertin, José; Romagny, Matthieu. Champs de Hurwitz. Mémoires de la Société Mathématique de France, Serie 2, no. 125-126 (2011), 219 p. doi : 10.24033/msmf.437. http://numdam.org/item/MSMF_2011_2_125-126__1_0/

[1] D. Abramovich, A. Corti & A. Vistoli« Twisted bundles and admissible covers », Comm. Algebra 31 (2003), p. 3547–3618. | MR | Zbl

[2] D. Abramovich & T. J. Jarvis« Moduli of twisted spin curves », Proc. Amer. Math. Soc. 131 (2003), p. 685–699. | MR | Zbl

[3] D. Abramovich & A. Vistoli« Compactifying the space of stable maps », J. Amer. Math. Soc. 15 (2002), p. 27–75. | MR | Zbl

[4] E. Arbarello & M. Cornalba« Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves », J. Algebraic Geom. 5 (1996), p. 705–749. | MR | Zbl

[5] A. Arsie & A. Vistoli« Stacks of cyclic covers of projective spaces », Compos. Math. 140 (2004), p. 647–666. | MR | Zbl

[6] M. Asada, M. Matsumoto & T. Oda« Local monodromy on the fundamental groups of algebraic curves along a degenerate stable curve », J. Pure Appl. Algebra 103 (1995), p. 235–283. | MR | Zbl

[7] H. Bass« Covering theory for graphs of groups », J. Pure Appl. Algebra 89 (1993), p. 3–47. | MR | Zbl

[8] A. Beauville« Prym varieties and the Schottky problem », Invent. Math. 41 (1977), p. 149–196. | MR | EuDML | Zbl

[9] S. Beckmann« Ramified primes in the field of moduli of branched coverings of curves », J. Algebra 125 (1989), p. 236–255. | MR | Zbl

[10] A. A. Beĭlinson & Y. I. Manin« The Mumford form and the Polyakov measure in string theory », Comm. Math. Phys. 107 (1986), p. 359–376. | MR | Zbl

[11] A. J. Bene« Combinatorial classes, hyperelliptic loci, and Hodge integrals », preprint arXiv :math.GT/0610603.

[12] D. J. BensonRepresentations and cohomology. I, Cambridge Studies in Advanced Math., vol. 30, Cambridge Univ. Press, 1991. | MR

[13] J. Bertin« Compactification des schémas de Hurwitz », C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), p. 1063–1066. | MR | Zbl

[14] J. Bertin & S. Maugeais« Déformations équivariantes des courbes semi-stables », Ann. Inst. Fourier (Grenoble) 55 (2005), p. 1905–1941. | MR | EuDML | Zbl | Numdam

[15] J. Bertin & A. Mézard« Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques », Invent. Math. 141 (2000), p. 195–238. | MR

[16] M. Boggi & M. Pikaart« Galois covers of moduli of curves », Compositio Math. 120 (2000), p. 171–191. | MR | Zbl

[17] J. Bryan, T. Graber & R. Pandharipande« The orbifold quantum cohomology of 2 / 3 and Hurwitz-Hodge integrals », J. Algebraic Geom. 17 (2008), p. 1–28. | MR | Zbl

[18] J.-L. Brylinski« Propriétés de ramification à l’infini du groupe modulaire de Teichmüller », Ann. Sci. École Norm. Sup. 12 (1979), p. 295–333. | MR | EuDML | Zbl | Numdam

[19] A. Cadoret & A. Tamagawa« Stratification of Hurwitz spaces by closed modular subvarieties », Pure Appl. Math. Q. 5 (2009), p. 227–253. | MR | Zbl

[20] D. Chen« Covers of the projective line and the moduli space of quadratic differentials », preprint arXiv :mathAG/1005.3120. | MR | Zbl

[21] C. Chevalley & A. Weil« Über das Verhalten der Integrale erster Gattung bei Automorphismen des Funktionenkörpers », Abh. Math. Sem. Hamburg Univ. 10 (1934), p. 358–361. | MR | JFM

[22] A. Chiodo« Towards an enumerative geometry of the moduli space of twisted curves and rth roots », Compos. Math. 144 (2008), p. 1461–1496. | MR | Zbl

[23] T. Coates & A. Givental« Quantum Riemann-Roch, Lefschetz and Serre », Ann. of Math. 165 (2007), p. 15–53. | MR | Zbl

[24] K. Coombes & D. Harbater« Hurwitz families and arithmetic Galois groups », Duke Math. J. 52 (1985), p. 821–839. | MR | Zbl

[25] A. F. Costa & S. M. Natanzon« Topological classification of p m actions on surfaces », Michigan Math. J. 50 (2002), p. 451–460. | MR | Zbl

[26] P. Dèbes« Arithmétique et espaces de modules de revêtements », in Number theory in progress, Vol. 1 (Zakopane-Kościelisko, 1997), de Gruyter, 1999, p. 75–102. | MR

[27] P. Dèbes & M. Emsalem« Harbater-Mumford components and towers of moduli spaces », J. Inst. Math. Jussieu 5 (2006), p. 351–371. | MR | Zbl

[28] P. Deligne« Le lemme de Gabber », Astérisque 127 (1985), p. 131–150. | MR | Numdam

[29] —, « Le déterminant de la cohomologie », in Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, Amer. Math. Soc., 1987, p. 93–177. | MR

[30] P. Deligne & D. Mumford« The irreducibility of the space of curves of given genus », Publ. Math. I.H.É.S. 36 (1969), p. 75–109. | MR | EuDML | Zbl | Numdam

[31] P. Deligne & M. Rapoport« Les schémas de modules de courbes elliptiques », in Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., vol. 349, Springer, 1973, p. 143–316. | MR | Zbl

[32] M. Demazure & P. GabrielGroupes algébriques, North-Holland, 1970.

[33] S. P. Diaz« On the Natanzon-Turaev compactification of the Hurwitz space », Proc. Amer. Math. Soc. 130 (2002), p. 613–618. | MR | Zbl

[34] S. P. Diaz & D. Edidin« Towards the homology of Hurwitz spaces », J. Differential Geom. 43 (1996), p. 66–98. | MR | Zbl

[35] R. Dijkgraaf« Mirror symmetry and elliptic curves », in The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkhäuser, 1995, p. 149–163. | MR | Zbl

[36] C. J. Earle« On the moduli of closed Riemann surfaces with symmetries », in Advances in the theory of riemann surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Princeton Univ. Press, 1971, p. 119–130. Ann. of Math. Studies, No. 66. | MR

[37] B. Edixhoven« Néron models and tame ramification », Compositio Math. 81 (1992), p. 291–306. | MR | EuDML | Zbl | Numdam

[38] A. L. Edmonds« Surface symmetry. I », Michigan Math. J. 29 (1982), p. 171–183. | MR | Zbl

[39] T. Ekedahl« Boundary behaviour of Hurwitz schemes », in The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkhäuser, 1995, p. 173–198. | MR | Zbl

[40] T. Ekedahl, S. Lando, M. Shapiro & A. Vainshtein« Hurwitz numbers and intersections on moduli spaces of curves », Invent. Math. 146 (2001), p. 297–327. | MR | Zbl

[41] G. Ellingsrud & K. Lønsted« An equivariant Lefschetz formula for finite reductive groups », Math. Ann. 251 (1980), p. 253–261. | MR | EuDML | Zbl

[42] M. Emsalem« Familles de revêtements de la droite projective », Bull. Soc. Math. France 123 (1995), p. 47–85. | MR | EuDML | Numdam

[43] A. Eskin, M. Kontsevich & A. Zorich« Lyapunov spectrum of square-tiled cyclic covers », J. Mod. Dyn. 5 (2011), p. 319–353. | MR | Zbl

[44] C. Faber & R. Pandharipande« Hodge integrals and Gromov-Witten theory », Invent. Math. 139 (2000), p. 173–199. | MR | Zbl

[45] G. Faltings« Moduli-stacks for bundles on semistable curves », Math. Ann. 304 (1996), p. 489–515. | MR | EuDML | Zbl

[46] M. D. Fried« Fields of definition of function fields and Hurwitz families—groups as Galois groups », Comm. Algebra 5 (1977), p. 17–82. | MR | Zbl

[47] —, « Introduction to modular towers : generalizing dihedral group–modular curve connections », in Recent developments in the inverse Galois problem (Seattle, WA, 1993), Contemp. Math., vol. 186, Amer. Math. Soc., 1995, p. 111–171. | MR | Zbl

[48] M. D. Fried & H. Völklein« The inverse Galois problem and rational points on moduli spaces », Math. Ann. 290 (1991), p. 771–800. | MR | EuDML | Zbl

[49] W. Fulton« Hurwitz schemes and irreducibility of moduli of algebraic curves », Ann. of Math. 90 (1969), p. 542–575. | MR | Zbl

[50] B. Van Geemen & F. Oort« A compactification of a fine moduli space of curves », in Resolution of singularities (Obergurgl, 1997), Progr. Math., vol. 181, Birkhäuser, 2000, p. 285–298. | MR | Zbl

[51] D. GiesekerLectures on moduli of curves, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 69, Published for the Tata Institute of Fundamental Research, Bombay, 1982. | MR | Zbl

[52] T. Graber & R. Vakil« Hodge integrals and Hurwitz numbers via virtual localization », Compositio Math. 135 (2003), p. 25–36. | MR | Zbl

[53] A. Grothendieck (éd.) – Revêtements étales et groupe fondamental, Lecture Notes in Math., vol. 224, Springer, 1971, Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1). | MR

[54] J. Harris, T. Graber & J. Starr« A note on Hurwitz schemes of covers of a positive genus curve », preprint arXiv :math.AG/0205056.

[55] J. Harris & I. MorrisonModuli of curves, Graduate Texts in Math., vol. 187, Springer, 1998. | MR | Zbl

[56] J. Harris & D. Mumford« On the Kodaira dimension of the moduli space of curves », Invent. Math. 67 (1982), p. 23–88. | MR | EuDML | Zbl

[57] R. HartshorneResidues and duality, Lecture Notes in Math., vol. 20, Springer, 1966, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. | MR | EuDML

[58] T. J. Jarvis« Torsion-free sheaves and moduli of generalized spin curves », Compositio Math. 110 (1998), p. 291–333. | MR | Zbl

[59] —, « Geometry of the moduli of higher spin curves », Internat. J. Math. 11 (2000), p. 637–663. | MR | Zbl

[60] —, « The Picard group of the moduli of higher spin curves », New York J. Math. 7 (2001), p. 23–47. | MR | EuDML | Zbl

[61] T. J. Jarvis, R. Kaufmann & T. Kimura« Pointed admissible G-covers and G-equivariant cohomological field theories », Compos. Math. 141 (2005), p. 926–978. | MR | Zbl

[62] —, « Stringy K-theory and the Chern character », Invent. Math. 168 (2007), p. 23–81. | MR | Zbl

[63] T. J. Jarvis & T. Kimura« Orbifold quantum cohomology of the classifying space of a finite group », in Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math., vol. 310, Amer. Math. Soc., 2002, p. 123–134. | MR | Zbl

[64] —, « A representation-valued relative Riemann-Hurwitz theorem and the Hurwitz-Hodge bundle », preprint arXiv :math.AG/08102488.

[65] E. Kani« Hurwitz spaces of genus 2 covers of an elliptic curve », Collect. Math. 54 (2003), p. 1–51. | MR | EuDML | Zbl

[66] G. Katz« How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties », Expo. Math. 21 (2003), p. 219–261. | MR | Zbl

[67] N. M. Katz & B. MazurArithmetic moduli of elliptic curves, Annals of Math. Studies, vol. 108, Princeton Univ. Press, 1985. | MR | Zbl

[68] F. F. Knudsen« The projectivity of the moduli space of stable curves. II. The stacks M g,n », Math. Scand. 52 (1983), p. 161–199. | MR | EuDML | Zbl

[69] F. F. Knudsen & D. Mumford« The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div” », Math. Scand. 39 (1976), p. 19–55. | MR | EuDML | Zbl

[70] A. Kokotov, D. Korotkin & P. Zograf« Isomonodromic tau function on the space of admissible covers », Adv. Math. 227 (2011), p. 586–600. | MR | Zbl

[71] M. Kontsevich« Enumeration of rational curves via torus actions », in The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkhäuser, 1995, p. 335–368. | MR | Zbl

[72] S. Lando« Ramified coverings of the two-dimensional sphere and intersection theory in spaces of meromorphic functions on algebraic curves », Uspekhi Mat. Nauk 57 (2002), p. 29–98. | MR

[73] G. Laumon & L. Moret-BaillyChamps algébriques, Ergebnisse Math. Grenzg., vol. 39, Springer, 2000. | MR

[74] E. Looijenga« Smooth Deligne-Mumford compactifications by means of Prym level structures », J. Algebraic Geom. 3 (1994), p. 283–293. | MR | Zbl

[75] —, « Correspondences between moduli spaces of curves », in Moduli of curves and abelian varieties, Aspects Math., E33, Vieweg, 1999, p. 131–150. | MR | Zbl

[76] Y. I. ManinFrobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society Colloquium Publications, vol. 47, Amer. Math. Soc., 1999. | MR | Zbl

[77] S. Mochizuki« The geometry of the compactification of the Hurwitz scheme », Publ. Res. Inst. Math. Sci. 31 (1995), p. 355–441. | MR | Zbl

[78] L. Moret-Bailly« Construction de revêtements de courbes pointées », J. Algebra 240 (2001), p. 505–534. | MR | Zbl

[79] I. Morrison & H. Pinkham« Galois Weierstrass points and Hurwitz characters », Ann. of Math. 124 (1986), p. 591–625. | MR | Zbl

[80] D. Mumford« Towards an enumerative geometry of the moduli space of curves », in Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser, 1983, p. 271–328. | MR

[81] D. Mumford, J. Fogarty & F. KirwanGeometric invariant theory, Ergebnisse Math. Grenzg., vol. 34, Springer, 1994. | MR | Zbl

[82] S. M. Natanzon« Moduli of Riemann surfaces, Hurwitz-type spaces, and their superanalogues », Uspekhi Mat. Nauk 54 (1999), p. 61–116. | MR | Zbl

[83] J. Nielsen« Die Struktur periodischer Transformationen von Flächen », Mat.-Fys. Medd. Danske Vid. Selsk. 15 (1937), p. 1–77 ; traduction anglaise : in Collected Papers 2, Birkhäuser (1986). | JFM

[84] A. Okounkov & R. Pandharipande« Gromov-Witten theory, Hurwitz numbers, and matrix models », in Algebraic geometry—Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., 2009, p. 325–414. | MR | Zbl

[85] M. Pikaart & A. J. De Jong« Moduli of curves with non-abelian level structure », in The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkhäuser, 1995, p. 483–509. | MR | Zbl

[86] M. Raynaud« p-groupes et réduction semi-stable des courbes », in The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser, 1990, p. 179–197. | MR

[87] —, « Spécialisation des revêtements en caractéristique p>0 », Ann. Sci. École Norm. Sup. 32 (1999), p. 87–126. | MR | EuDML

[88] M. Romagny« Sur quelques aspects des champs de revêtements de courbes algébriques », thèse de doctorat, Université Grenoble 1, 2002.

[89] —, « The stack of Potts curves and its fibre at a prime of wild ramification », J. Algebra 274 (2004), p. 772–803. | MR | Zbl

[90] —, « Group actions on stacks and applications », Michigan Math. J. 53 (2005), p. 209–236. | MR | Zbl

[91] M. Saïdi« Revêtements modérés et groupe fondamental de graphe de groupes », Compositio Math. 107 (1997), p. 319–338. | MR

[92] J-P. SerreReprésentations linéaires des groupes finis, 5e éd., Hermann, 1988. | MR

[93] —, Topics in Galois theory, second éd., Research Notes in Math., vol. 1, A K Peters Ltd., 2008.

[94] C. S. Seshadri« Geometric reductivity over arbitrary base », Advances in Math. 26 (1977), p. 225–274. | MR | Zbl

[95] V. P. SnaithExplicit Brauer induction, Cambridge Studies in Advanced Math., vol. 40, Cambridge Univ. Press, 1994. | MR | Zbl

[96] H.-H. Tseng« Orbifold quantum Riemann-Roch, Lefschetz and Serre », Geom. Topol. 14 (2010), p. 1–81. | MR | Zbl

[97] A. Vistoli« Intersection theory on algebraic stacks and on their moduli spaces », Invent. Math. 97 (1989), p. 613–670. | MR | EuDML | Zbl

[98] B. Wajnryb« Orbits of Hurwitz action for coverings of a sphere with two special fibers », Indag. Math. (N.S.) 7 (1996), p. 549–558. | MR | Zbl

[99] S. Wewers« Construction of Hurwitz spaces », Thèse, Universität Duisburg-Essen, 1998. | Zbl

[100] —, « Deformation of tame admissible covers of curves », in Aspects of Galois theory (Gainesville, FL, 1996), London Math. Soc. Lecture Note Ser., vol. 256, Cambridge Univ. Press, 1999, p. 239–282. | MR | Zbl

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