×

Irreducibility of the space of dihedral covers of the projective line of a given numerical type. (English) Zbl 1268.14025

From the abstract: We show in this paper that the set of irreducible components of the family of Galois coverings of \(\mathbb P^1(\mathbb C)\) with Galois group isomorphic to \(D_n\) is in bijection with the set of possible numerical types.

MSC:

14H10 Families, moduli of curves (algebraic)
14H30 Coverings of curves, fundamental group
57M12 Low-dimensional topology of special (e.g., branched) coverings
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] I. Bauer - F. Catanese, Generic lemniscates of algebraic functions. Math. Ann. 307, no. 3, 417-444 (1997). · Zbl 0873.57004 · doi:10.1007/s002080050042
[2] R . B i g ge r s - M . Fri ed, Irreducibility of moduli spaces of cyclic unramified covers of genus g curves. Trans. Am. Math. Soc. 295, 59-70 (1986). · Zbl 0601.14022 · doi:10.2307/2000145
[3] F. C at ane s e , Moduli of algebraic surfaces. Theory of moduli (Montecatini Terme, 1985), 1-83, Lecture Notes in Math., 1337, Springer, Berlin, (1988). · Zbl 0658.14017
[4] F . C a tan es e, Fibred Surfaces, varieties isogenous to a product and related moduli spaces. Amer. J. Math. 122 (2000), no. 1, 1-44. · Zbl 0983.14013 · doi:10.1353/ajm.2000.0002
[5] F. Cata ne se, Di\?erentiable and deformation type of algebraic surfaces, real and symplectic structures. Symplectic 4-manifolds and algebraic surfaces, 55-167, Lecture Notes in Math., 1938, Springer, Berlin, (2008). · Zbl 1145.14001
[6] F . C a t a n es e, Irreducibility of the space of cyclic covers of algebraic curves of fixed numerical type and the irreducible components of Sing\eth MgÞ. To appear in the Proceedings volume of “The Conference on Geometry” honoring Shing- Tung Yau’s 60th birthday. “Advanced Lectures in Mathematics” series of Inter- national Press, in cooperation with the Higher Education Press and the Stefan Banach International Mathematical Centre (Institute of Mathematics Polish Academy of Sciences Publishing House). arXiv:1011.0316.
[7] A . C l ebs c h, Zur Theorie der Riemann’schen Fla \"chen. Math. Ann. 6, 216-230 (1872).
[8] A . Com ess atti , Sulle superficie multiple cicliche. Rendiconti Seminario Padova 1, 1-45 (1930).
[9] M. Cornalba, On the locus of curves with automorphisms. Ann. Mat. Pura Appl., IV. Ser. 149, 135-151 (1987). · Zbl 0649.14013 · doi:10.1007/BF01773930
[10] M. Cornalba, Erratum: On the locus of curves with automorphisms. Ann. Mat. Pura Appl. (4) 187, No. 1, 185-186 (2008). · Zbl 1150.14003 · doi:10.1007/s10231-006-0031-0
[11] N. M. Dunfield - W. P. Thurston, Finite covers of random 3-manifolds. Invent. Math. 166, no. 3, 457-521 (2006). · Zbl 1111.57013 · doi:10.1007/s00222-006-0001-6
[12] A . L. E dm o nd s, Surface symmetry I. Michigan Math. J. 29, no. 2, 171-183 (1982).
[13] A . L. Ed mo nd s, Surface symmetry II. Michigan Math. J. 30, no. 2, 143-154 (1983).
[14] W. Fulton , Hurwitz schemes and irreducibility of moduli of algebraic curves. Ann. of Math. (2) 90, 542-575 (1969). · Zbl 0194.21901 · doi:10.2307/1970748
[15] T. Graber - J. H arr is - J. Sta rr , A note on Hurwitz schemes of covers of a positive genus curve. arXiv: math. AG/0205056.
[16] A . H u r w i tz, Ueber Riemann’schen Fla \"chen mit gegebenen Verzweigung- spunkten. Math. Ann. 39, 1-61 (1891).
[17] V. Kanev, Hurwitz spaces of Galois coverings of P1, whose Galois groups are Weyl groups, J. Algebra 305 (2006) 442-456. · Zbl 1118.14034 · doi:10.1016/j.jalgebra.2006.01.008
[18] V. Kanev, Irreducibility of Hurwitz spaces. arXiv: math. AG/0509154.
[19] P . Kl ui tma n n , Hurwitz action and finite quotients of braid groups. In: Braids (Santa Cruz, CA 1986). Contemporary Mathematics, vol. 78, pp. 299-325. AMS, Providence (1988). · Zbl 0701.20019
[20] R . P a r d i n i, Abelian covers of algebraic varieties. J. Reine Angew. Math. 417 (1991), 191-213. 309 · Zbl 0721.14009 · doi:10.1515/crll.1991.417.191
[21] C. Sia, Hurwitz equivalence in tuples of dihedral groups, dicyclic groups, and semidihedral groups. Electron. J. Combin. 16, no. 1, Research Paper 95, 17 pp (2009). · Zbl 1191.20035
[22] F . V et r o, Irreducibility of Hurwitz spaces of coverings with one special fiber. Indag. Math. (N.S.) 17, no. 1, 115-127 (2006). · Zbl 1101.14040 · doi:10.1016/S0019-3577(06)80010-8
[23] F . Ve tro , Irreducibility of Hurwitz spaces of coverings with monodromy groups Weyl groups of type W \eth Bd Þ. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10, no. 2, 405-431 (2007). · Zbl 1178.14029
[24] F . V et r o, Irreducibility of Hurwitz spaces of coverings with one special fiber and monodromy group a Weyl group of type Dd . Manuscripta Math. 125, no. 3, 353-368 (2008). · Zbl 1139.14023 · doi:10.1007/s00229-007-0153-8
[25] B. Wajnryb , Orbits of Hurwitz action for coverings of a sphere with two special fibers. Indag. Math. (N.S.) 7, no. 4, 549-558 (1996). · Zbl 0881.57001 · doi:10.1016/S0019-3577(97)89139-2
[26] B. Wajnryb, An elementary approach to the mapping class group of a surface. Geom. Topol. 3, 405-466 (1999). · Zbl 0947.57015 · doi:10.2140/gt.1999.3.405
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.