Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

Abstract
Given a smooth, projective curve \(Y\), a finite group \(G\) and a positive integer $n$ we study smooth, proper families \(X\to Y\times S\to S\) of Galois covers of \(Y\) with Galois group isomorphic to $G$ branched in \(n\) points, parameterized by algebraic varieties \(S\). When \(G\) is with trivial center we prove that the Hurwitz space \(H^G_n(Y)\) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary \(G\) we prove that \(H^G_n(Y)\) is a coarse moduli variety. For families of pointed Galois covers of \((Y,y_0)\) we prove that the Hurwitz space \(H^G_n(Y,y_0)\) is a fine moduli variety, and construct explicitly the universal family, for arbitrary group \(G\). We use classical tools of algebraic topology and of complex algebraic geometry.
2020 Mathematics Subject Classification:
14H30, 14H10, 14D22Keywords
Galois cover of a curve, family of covers, Hurwitz space, moduli space
Author Details
Vassil Kanev
Dipartimento di Matematica e Informatica
Università di Palermo
Via Archirafi, 34
90123 Palermo, Italy
e-mail: vassil.kanev@unipa.it
References
- D. Abramovich, A. Corti, A. Vistoli. Twisted bundles and admissible covers. Comm. Algebra 31, 8 (2003) 3547–3618.
- V. Alexeev, R. Donagi, G. Farkas, E. Izadi A. Ortega. The uniformization of the moduli space of principally polarized abelian 6-folds. J. Reine Angew. Math. 761 (2020), 163–217.
- A. Altman, S. Kleiman. Introduction to Grothendieck duality theory. Lecture Notes in Math., vol. 146. Berlin-New York, Springer-Verlag, 1970.
- J. Bertin, M. Romagny. Champs de Hurwitz. Mém. Soc. Math. Fr. (N.S.) no. 125–126, 2011, 219 pp.
- J. S. Birman. Braids, links, and mapping class groups. Ann. of Math. Stud., No. 82, Princeton, N.J., Princeton University Press; Tokyo, University of Tokyo Press, 1974.
- F. A. Bogomolov, V. S. Kulikov. The ambiguity index of an equipped finite group. Eur. J. Math. 1, 2 (2015), 260–278.
- A. Carocca, H. Lange, R. E. Rodríguez, A. M. Rojas. Prym-Tyurin varieties via Hecke algebras. J. Reine Angew. Math. 634 (2009), 209–234.
- F. Catanese, M. Lönne, F. Perroni. Irreducibility of the space of dihedral covers of the projective line of a given numerical type. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22, 3 (2011), 291–309.
- P. Dèbes. Arithmétique et espaces de modules de revêtements. [Arithmetic and moduli spaces of covers] In: Number theory in progress, vol. 1 (Eds K. Győry, H. Iwaniec and J. Urbanowicz), Berlin, de Gruyter, 1999, 75–102.
- R. Donagi. Decomposition of spectral covers. In: Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). Astérisque, no. 218 (1993), 145–175.
- M. Emsalem. Familles de revêtements de la droite projective [Families of coverings of the projective line]. Bull. Soc. Math. France 123, 1 (1995), 47–85.
- M. Emsalem. Espaces de Hurwitz. Sémin. Congr., vol. 5, 2001, 63–99, Soc. Math. France, Paris.
- E. Fadell, L. Neuwirth. Configuration spaces. Math. Scand. 10 (1962), 111–118.
- B. Fantechi, L. Göttsche. Local properties and Hilbert schemes of points, In: Fundamental algebraic geometry, 139–178. Math. Surveys Monogr., vol. 123, Providence, RI, Amer. Math. Soc., 2005.
- G. Fischer. Complex analytic geometry. Lecture Notes in Math., vol. 538. Berlin-New York, Springer-Verlag, 1976.
- O. Forster. Lectures on Riemann surfaces. Grad. Texts in Math., vol. 81. New York-Berlin, Springer-Verlag, 1981.
- M. Fried. Fields of definition of function fields and Hurwitz families—groups as Galois groups. Comm. Algebra 5, 1 (1977), 17–82.
- M. D. Fried, H. Völklein. The inverse Galois problem and rational points on moduli spaces. Math. Ann. 290, 4 (1991), 771–800.
- W. Fulton. Hurwitz schemes and irreducibility of moduli of algebraic curves. Ann. of Math. (2) 90 (1969), 542–575.
- U. Görtz, T. Wedhorn. Algebraic geometry I: Schemes with examples and exercises. Adv. Lectures Math. Wiesbaden, Vieweg + Teubner, 2010.
- H. Grauert, R. Remmert. Coherent analytic sheaves. Grundlehren Math. Wiss., vol. 265 [Fundamental Principles of Mathematical Sciences]. Berlin, Springer-Verlag, 1984.
- A. Grothendieck. Éléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents I. Inst. Hautes Études Sci. Publ. Math. (1961), no. 11, 167 pp.
- A. Grothendieck. Techniques de construction en g´eom´etrie analytique. VI. Étude locale des morphismes: germes d’espaces analytiques, platitude, morphismes simples. In: Séminaire Henri Cartan, Tome 13 (1960–1961) no. 1, Exposé no. 13, 13 pp.
- A. Grothendieck. Techniques de construction en géométrie analytique. VII. Étude locale des morphismes: éléments de calcul infinitésimal. In: Séminaire Henri Cartan, Tome 13 (1960–1961) no. 2, Exposé no. 14, 27 pp.
- J. Harris. Algebraic geometry: a first course. Grad. Texts in Math., vol. 133. New York, Springer-Verlag, 1992.
- R. Hartshorne. Algebraic geometry, Grad. Texts in Math., vol. 52, New York-Heidelberg, Springer-Verlag, 1977.
- S. T. Hu. Homotopy theory. Pure Appl. Math., vol. VIII, New York-London, Academic Press, 1959.
- V. Kanev. Spectral curves and Prym-Tjurin varieties. I. In: Abelian varieties (Egloffstein, 1993) (Eds W. Barth, K. Hulek and H. Lange) Berlin, de Gruyter, 1995, 151–198.
- V. Kanev. Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of abelian threefolds. Ann. Mat. Pura Appl. (4) 183, 3 (2004), 333–374.
- V. Kanev. Hurwitz spaces of quadruple coverings of elliptic curves and the moduli space of abelian threefolds A3(1, 1, 4). Math. Nachr. 278, 1–2 (2005), 154–172.
- V. Kanev. Hurwitz spaces of Galois coverings of P1, whose Galois groups are Weyl groups. J. Algebra 305, 1 (2006), 442–456.
- V. Kanev. Irreducible components of Hurwitz spaces parameterizing Galois coverings of curves of positive genus, Pure Appl. Math. Q. 10, 2 (2014), 193–222.
- V. Kanev. A criterion for extending morphisms from open subsets of smooth fibrations of algebraic varieties. J. Pure Appl. Algebra 225, 4 (2021), no. 4, Paper No. 106553, 10 pp.
- V. Kanev. Hurwitz moduli varieties parameterizing pointed covers of an algebraic curve with a fixed monodromy group, 2024, https://arxiv.org/abs/2403.12756.
- N. M. Katz, B. Mazur. Arithmetic moduli of elliptic curves. Ann. of Math. Stud., vol. 108. Princeton, NJ, Princeton University Press, 1985
- V. S. Kulikov. Factorization semigroups and irreducible components of the Hurwitz space. Izv. Ross. Akad. Nauk Ser. Mat. 75, 4 (2011), 49–90 (in Russian); English translation in Izv. Math. 75, 4 (2011), 711–748.
- V. S. Kulikov. Factorizations in finite groups, Mat. Sb. 204, 2 (2013), 87–116 (in Russian); English translation in Sb. Math. 204, 1–2 (2013), 237–263.
- V. S. Kulikov, V. M. Kharlamov. The semigroups of coverings. Izv. Ross. Akad. Nauk Ser. Mat. 77, 3 (2013), 163–198 (in Russian); English translation in Izv. Math. 77, 3 (2013), 594–626.
- Q. Liu. Algebraic geometry and arithmetic curves. Oxf. Grad. Texts Math., vol. 6. Oxford Sci. Publ. Oxford, Oxford University Press, 2002.
- W. S. Massey. A basic course in algebraic topology. Grad. Texts in Math., vol. 127, New York, Springer-Verlag, 1991.
- H. Matsumura. Commutative algebra, 2nd edt. Math. Lecture Note Ser., vol. 56. Reading, MA, Benjamin/Cummings Publishing Co., Inc., 1980.
- H. Matsumura. Commutative ring theory. Cambridge Stud. Adv. Math., vol. 8. Cambridge, Cambridge University Press, 1986.
- D. Mumford. Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 34, Berlin-New York, Springer-Verlag, 1965.
- D. Mumford. Lectures on curves on an algebraic surface. With a section by G. M. Bergman. Ann. of Math. Stud., No. 59. Princeton, NJ, Princeton University Press, 1966.
- R. Narasimhan. Several complex variables. Chicago Lectures in Math., Chicago, IL, University of Chicago Press, 1971.
- T. Peternell, R. Remmert. Differential calculus, holomorphic maps and linear structures on complex spaces. In: Several complex variables, VII (Eds H. Grauert, Th. Peternell and R. Remmert). Encyclopaedia Math. Sci., vol. 74. Berlin, Springer, 1994, 97–144.
- M. Raynaud. Géométrie algébrique et géométrie analytique, Exposé XII (from unpublished notes by A. Grothendieck), In: Revêtements étales et groupe fondamental: Séminaire de Géométrie Algébrique (SGA 1), Lecture Notes in Mathematics, vol. 224, Springer-Verlag, Berlin-New York, 1971, 311–343. (Updated edition available at arXiv:math/0206203 [math.AG]).
- M. Romagny, S. Wewers. Hurwitz spaces. In: Groupes de Galois arithmétiques et différentiels (Eds D. Bertrand and P. Dèbes), Sémin. Congr., vol. 13. Paris, Soc. Math. France, 2006, 313–341.
- E. Sernesi. Deformations of algebraic schemes. Grundlehren der mathematischen Wissenschaften, vol. 334 [Fundamental Principles of Mathematical Sciences]. Berlin, Springer-Verlag, 2006.
- J.-P. Serre. Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier (Grenoble) 6 (1955/56), 1–42.
- J.-P. Serre. Espaces fibrés algébriques. In: Séminaire Claude Chevalley, Tome 3 (1958), Exposé no. 1, 37 pp, http://www.numdam.org/item/SCC_1958__3__A1_0/.
- J.-P. Serre. Algebraic groups and class fields. Grad. Texts in Math., vol. 117. New York, Springer-Verlag, 1988.
- I. R. Shafarevich. Basic algebraic geometry 2: Schemes and complex manifolds. Berlin, Springer-Verlag, 1994.
- E. H. Spanier. Algebraic topology. New York-Toronto, Ont.-London, McGraw-Hill Book Co., 1966.
- F. Vetro. Coverings with special fibers and the monodromy group Sd. Izv. Ross. Akad. Nauk Ser. Mat. 76, 6 (2012), 39–44 (in Russian); English translation in Izv. Math. 76, 6 (2012), 1110–1115.
- H. Völklein. Moduli spaces for covers of the Riemann sphere. Israel J. Math. 85, 1–3 (1994), 407–430.
- H. Völklein. Groups as Galois groups: An introduction. Cambridge Stud. Adv. Math., vol. 53. Cambridge, Cambridge University Press, 1996.
- S.Wewers. Construction of Hurwitz spaces, Ph. D. Thesis, Preprint No. 21, 1988, Institut für Experimentelle Mathematik, Universität GH Essen.