Skip to main content

Kummer covers and braid monodromy

  • Enrique Artal Bartolo (a1), José Ignacio Cogolludo-Agustín (a1) and Jorge Ortigas-Galindo (a2)

In this work, we describe a method to construct the generic braid monodromy of the preimage of a curve by a Kummer cover. This method is interesting since it combines two techniques, namely, the construction of a highly non-generic braid monodromy and a systematic method to go from a non-generic to a generic braid monodromy. The latter process, called generification, is independent from Kummer covers, and it can be applied in more general circumstances since non-generic braid monodromies appear more naturally and are oftentimes much easier to compute. Explicit examples are computed using these techniques.

Hide All
1. Amram, M., Ciliberto, C., Miranda, R. and Teicher, M., Braid monodromy factorization for a non-prime $K 3$ surface branch curve, Israel J. Math. 170 (2009), 6193.
2. Amram, M. and Teicher, M., On the degeneration, regeneration and braid monodromy of $T\times T$ , Acta Appl. Math. 75 (1–3) (2003), 195270 Monodromy and differential equations (Moscow, 2001).
3. Artal, E., Sur les couples de Zariski, J. Algebraic Geom. 3 (2) (1994), 223247.
4. Artal, E. and Carmona, J., Zariski pairs, fundamental groups and Alexander polynomials, J. Math. Soc. Japan 50 (3) (1998), 521543.
5. Artal, E., Carmona, J. and Cogolludo-Agustín, J. I., Braid monodromy and topology of plane curves, Duke Math. J. 118 (2) (2003), 261278.
6. Artal, E., Carmona, J., Cogolludo-Agustín, J. I. and Tokunaga, H., Sextics with singular points in special position, J. Knot Theory Ramifications 10 (4) (2001), 547578.
7. Artal, E., Cogolludo-Agustín, J. I. and Tokunaga, H., A survey on Zariski pairs, in Algebraic geometry in East Asia—Hanoi 2005, Adv. Stud. Pure Math., Volume 50, pp. 1100 (Math. Soc. Japan, Tokyo, 2008).
8. Arvola, W. A., The fundamental group of the complement of an arrangement of complex hyperplanes, Topology 31 (4) (1992), 757765.
9. Auroux, D. and Katzarkov, L., Branched coverings of $\mathbf{C} {\mathrm{P} }^{2} $ and invariants of symplectic 4-manifolds, Invent. Math. 142 (3) (2000), 631673.
10. Barthel, G., Hirzebruch, F. and Höfer, T., Geradenkonfigurationen und Algebraische Flächen, Aspects of Mathematics, D4 (Friedr. Vieweg & Sohn, Braunschweig, 1987).
11. Bessis, D. and Michel, J., Explicit presentations for exceptional braid groups, Experiment. Math. 13 (3) (2004), 257266.
12. Bigelow, S. J., Braid groups are linear, J. Amer. Math. Soc. 14 (2) (2001), 471486 electronic.
13. Carmona, J., Monodromía de trenzas de curvas algebraicas planas, Ph.D. thesis, Universidad de Zaragoza, 2003.
14. Catanese, F., Lönne, M. and Wajnryb, B., Moduli spaces of surfaces and monodromy invariants, in Proceedings of the Gökova Geometry-Topology Conference 2009, pp. 5898 (Int. Press, Somerville, MA, 2010).
15. Catanese, F. and Wajnryb, B., The 3-cuspidal quartic and braid monodromy of degree 4 coverings, in Projective varieties with unexpected properties, pp. 113129 (Walter de Gruyter GmbH & Co. KG, Berlin, 2005).
16. Chisini, O., Una suggestiva rappresentazione reale per le curve algebriche piane, Ist. Lombardo, Rend., II. Ser. 66 (1933), 11411155.
17. Cogolludo-Agustín, J. I., Fundamental group for some cuspidal curves, Bull. London Math. Soc. 31 (2) (1999), 136142.
18. Cogolludo-Agustín, J. I., Braid monodromy of algebraic curves, Ann. Math. Blaise Pascal 18 (1) (2011), 141209.
19. Cogolludo-Agustín, J. I. and Kloosterman, R., Mordell–Weil groups and Zariski triples, in Geometry and Arithmetic, EMS Ser. Congr. Rep., pp. 7589 (Eur. Math. Soc., Zürich, 2012).
20. Cohen, D. C. and Suciu, A. I., The braid monodromy of plane algebraic curves and hyperplane arrangements, Comment. Math. Helv. 72 (2) (1997), 285315.
21. Cordovil, R. and Fachada, J. L., Braid monodromy groups of wiring diagrams, Boll. Unione Mat. Ital. B (7) 9 (2) (1995), 399416.
22. Degtyarëv, A. I., On deformations of singular plane sextics, J. Algebraic Geom. 17 (1) (2008), 101135.
23. Dolgachev, I. and Libgober, A., On the fundamental group of the complement to a discriminant variety, in Algebraic geometry (Chicago, Ill., 1980), Lecture Notes in Mathematics, Volume 862, pp. 125 (Springer, Berlin, 1981).
24. The GAP Group, GAP – groups, algorithms, and programming, version 4.4, 2004, available at (
25. Hirano, A., Construction of plane curves with cusps, Saitama Math. J. 10 (1992), 2124.
26. van Kampen, E. R., On the fundamental group of an algebraic curve, Amer. J. Math. 55 (1933), 255260.
27. Krammer, D., The braid group ${B}_{4} $ is linear, Invent. Math. 142 (3) (2000), 451486.
28. Kulikov, Vik. S., Generic coverings of the plane and braid monodromy invariants, in The Fano Conference, pp. 533558 (Univ. Torino, Turin, 2004).
29. Kulikov, Vik. S. and Teicher, M., Braid monodromy factorizations and diffeomorphism types, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2) (2000), 89120.
30. Lawrence, R. J., Homological representations of the Hecke algebra, Comm. Math. Phys. 135 (1) (1990), 141191.
31. Libgober, A., On the Poincaré group of rational plane curves, Amer. J. Math. 58 (1936), 607619.
32. Libgober, A., The topological discriminant group of a Riemann surface of genus $p$ , Amer. J. Math. 59 (1937), 335358.
33. Libgober, A., On the homotopy type of the complement to plane algebraic curves, J. Reine Angew. Math. 367 (1986), 103114.
34. Libgober, A., Invariants of plane algebraic curves via representations of the braid groups, Invent. Math. 95 (1) (1989), 2530.
35. Lönne, M., Fundamental groups of projective discriminant complements, Duke Math. J. 150 (2) (2009), 357405.
36. MacLane, S., Some interpretations of abstract linear dependence in terms of projective geometry, Amer. J. Math. 58 (1) (1936), 236240.
37. Moishezon, B. G., Stable branch curves and braid monodromies, in Algebraic geometry (Chicago, Ill., 1980), Lecture Notes in Mathematics, Volume 862, pp. 107192 (Springer, Berlin, 1981).
38. Moishezon, B. G. and Teicher, M., Braid group techniques in complex geometry. IV. Braid monodromy of the branch curve ${S}_{3} $ of ${V}_{3} \rightarrow \mathbf{C} {\mathrm{P} }^{2} $ and application to ${\pi }_{1} (\mathbf{C} {\mathrm{P} }^{2} - {S}_{3} , \ast )$ , in Classification of algebraic varieties (L’Aquila, 1992), Contemp. Math., Volume 162, pp. 333358 (Amer. Math. Soc, Providence, RI, 1994).
39. Rudolph, L., Algebraic functions and closed braids, Topology 22 (2) (1983), 191202.
40. Salvetti, M., Arrangements of lines and monodromy of plane curves, Compositio Math. 68 (1) (1988), 103122.
41. Uludağ, A. M., More Zariski pairs and finite fundamental groups of curve complements, Manuscripta Math. 106 (3) (2001), 271277.
42. Zariski, O., On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. Math. 51 (1929), 305328.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 13 *
Loading metrics...

Abstract views

Total abstract views: 118 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 18th March 2018. This data will be updated every 24 hours.