Geodesic
Covering semigroups
Izvestiya. Mathematics , Tome 77 (2013) no. 3, pp. 594-626.

Voir la notice de l'article provenant de la source Math-Net.Ru

We introduce and study a semigroup structure on the set of irreducible components of the Hurwitz space of marked coverings of a complex projective curve with given Galois group of the coverings and fixed ramification type. As an application, we give new conditions on the ramification type that are sufficient for the irreducibility of the Hurwitz spaces, suggest some bounds on the number of irreducible components under certain more general conditions, and show that the number of irreducible components coincides with the number of topological classes of the coverings if the number of branch points is big enough.
Keywords: irreducible components of the Hurwitz space of finite-sheeted coverings of projective curves, semigroups over groups. 
@article{IM2_2013_77_3_a9,
     author = {Vik. S. Kulikov and V. M. Kharlamov},
     title = {Covering semigroups},
     journal = {Izvestiya. Mathematics },
     pages = {594--626},
     publisher = {mathdoc},
     volume = {77},
     number = {3},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2013_77_3_a9/}
}
TY  - JOUR
AU  - Vik. S. Kulikov
AU  - V. M. Kharlamov
TI  - Covering semigroups
JO  - Izvestiya. Mathematics 
PY  - 2013
SP  - 594
EP  - 626
VL  - 77
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2013_77_3_a9/
LA  - en
ID  - IM2_2013_77_3_a9
ER  - 
%0 Journal Article
%A Vik. S. Kulikov
%A V. M. Kharlamov
%T Covering semigroups
%J Izvestiya. Mathematics 
%D 2013
%P 594-626
%V 77
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2013_77_3_a9/
%G en
%F IM2_2013_77_3_a9
Vik. S. Kulikov; V. M. Kharlamov. Covering semigroups. Izvestiya. Mathematics , Tome 77 (2013) no. 3, pp. 594-626. http://geodesic.mathdoc.fr/item/IM2_2013_77_3_a9/

[1] A. Clebsch, “Zur Theorie der Riemann'schen Fläche”, Math. Ann., 6:2 (1873), 216–230 | DOI | MR | Zbl

[2] A. Hurwitz, “Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten”, Math. Ann., 39:1 (1891), 1–61 | DOI | MR | Zbl

[3] I. Berstein, A. L. Edmonds, “On the classification of generic branched coverings of surfaces”, Illinois J. Math., 28:1 (1984), 64–82 | MR | Zbl

[4] B. Wajnryb, “Orbits of Hurwitz action for coverings of a sphere with two special fibers”, Indag. Math. (N.S.), 7:4 (1996), 549–558 | DOI | MR | Zbl

[5] A. N. Protopopov, “Topological classification of branched coverings of the two-dimensional sphere”, J. Soviet Math., 52:1 (1990), 2832–2846 | DOI | MR | Zbl | Zbl

[6] M. D. Fried, H. Völklein, “The inverse Galois problem and rational points on moduli spaces”, Math. Ann., 290:1 (1991), 771–800 | DOI | MR | Zbl

[7] Vik. S. Kulikov, “Factorization semigroups and irreducible components of the Hurwitz space”, Izv. Math., 75:4 (2011), 711–748 | DOI | DOI | MR | Zbl

[8] Vik. S. Kulikov, “Factorizations in finite groups”, Sb. Math., 204:2 (2013), 237–263 | DOI | DOI

[9] S. M. Natanzon, “Spaces of real meromorphic functions on real algebraic curves”, Soviet Math. Dokl., 30:3 (1984), 724–726 | MR | Zbl

[10] Vik. S. Kulikov, “Factorization semigroups and irreducible components of the Hurwitz space. II”, Izv. Math., 76:2 (2012), 356–364 | DOI | DOI | MR | Zbl

[11] T. Graber, J. Harris, J. Starr, A note on Hurwitz schemes of covers of a positive genus curve, arXiv: math/0205056

[12] V. Kanev, Irreducibility of Hurwitz spaces, arXiv: math/0509154

[13] F. Vetro, “Irreducibility of Hurwitz spaces of coverings with one special fiber”, Indag. Math. (N.S.), 17:1 (2006), 115–127 | DOI | MR | Zbl

[14] F. Vetro, “A note on coverings with special fibres and monodromy group $S_{d}$”, Izv. Math., 76:6 (2012), 1110–1115 | DOI | DOI | Zbl

[15] Vik. S. Kulikov, “Hurwitz curves”, Russian Math. Surveys, 62:6 (2007), 1043–1119 | DOI | DOI | MR | Zbl

[16] Vik. S. Kulikov, V. M. Kharlamov, “On braid monodromy factorizations”, Izv. Math., 67:3 (2003), 499–534 | DOI | DOI | MR | Zbl

[17] R. H. Fox, “Covering spaces with singularities”, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton Univ. Press, Princeton, NJ, 1957, 243–257 | MR | Zbl

[18] E. Fadell, L. Neuwirth, “Configuration spaces”, Math. Scand., 10 (1962), 111–118 | MR | Zbl

[19] R. H. Fox, L. Neuwirth, “The braid groups”, Math. Scand., 10 (1962), 119–126 | MR | Zbl

[20] W. Fulton, “Hurwitz schemes and irreducibility of moduli of algebraic curves”, Ann. of Math. (2), 90:3 (1969), 542–575 | DOI | MR | Zbl