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
Keywords: Algebraic curve, covering, Galois covering, algebraic stack, compactification, level structure, Riemann-Hurwitz, cyclic covering, tautological classes
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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/

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