Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-19T12:28:09.886Z Has data issue: false hasContentIssue false

Kummer covers and braid monodromy

Published online by Cambridge University Press:  17 October 2013

Enrique Artal Bartolo
Affiliation:
Departamento de Matemáticas, IUMA, Universidad de Zaragoza, C. Pedro Cerbuna 12, 50009 Zaragoza, Spain (artal@unizar.es; jicogo@unizar.es)
José Ignacio Cogolludo-Agustín
Affiliation:
Departamento de Matemáticas, IUMA, Universidad de Zaragoza, C. Pedro Cerbuna 12, 50009 Zaragoza, Spain (artal@unizar.es; jicogo@unizar.es)
Jorge Ortigas-Galindo
Affiliation:
Centro Universitario de la Defensa-IUMA, Academia General Militar, Ctra. de Huesca s/n., 50090, Zaragoza, Spain (jortigas@unizar.es)

Abstract

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.

Type
Research Article
Copyright
©Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
Artal, E., Sur les couples de Zariski, J. Algebraic Geom. 3 (2) (1994), 223247.Google Scholar
Artal, E. and Carmona, J., Zariski pairs, fundamental groups and Alexander polynomials, J. Math. Soc. Japan 50 (3) (1998), 521543.Google Scholar
Artal, E., Carmona, J. and Cogolludo-Agustín, J. I., Braid monodromy and topology of plane curves, Duke Math. J. 118 (2) (2003), 261278.Google Scholar
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.CrossRefGoogle Scholar
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).Google Scholar
Arvola, W. A., The fundamental group of the complement of an arrangement of complex hyperplanes, Topology 31 (4) (1992), 757765.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Barthel, G., Hirzebruch, F. and Höfer, T., Geradenkonfigurationen und Algebraische Flächen, Aspects of Mathematics, D4 (Friedr. Vieweg & Sohn, Braunschweig, 1987).CrossRefGoogle Scholar
Bessis, D. and Michel, J., Explicit presentations for exceptional braid groups, Experiment. Math. 13 (3) (2004), 257266.CrossRefGoogle Scholar
Bigelow, S. J., Braid groups are linear, J. Amer. Math. Soc. 14 (2) (2001), 471486 electronic.CrossRefGoogle Scholar
Carmona, J., Monodromía de trenzas de curvas algebraicas planas, Ph.D. thesis, Universidad de Zaragoza, 2003.Google Scholar
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).Google Scholar
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).CrossRefGoogle Scholar
Chisini, O., Una suggestiva rappresentazione reale per le curve algebriche piane, Ist. Lombardo, Rend., II. Ser. 66 (1933), 11411155.Google Scholar
Cogolludo-Agustín, J. I., Fundamental group for some cuspidal curves, Bull. London Math. Soc. 31 (2) (1999), 136142.CrossRefGoogle Scholar
Cogolludo-Agustín, J. I., Braid monodromy of algebraic curves, Ann. Math. Blaise Pascal 18 (1) (2011), 141209.CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
Cohen, D. C. and Suciu, A. I., The braid monodromy of plane algebraic curves and hyperplane arrangements, Comment. Math. Helv. 72 (2) (1997), 285315.CrossRefGoogle Scholar
Cordovil, R. and Fachada, J. L., Braid monodromy groups of wiring diagrams, Boll. Unione Mat. Ital. B (7) 9 (2) (1995), 399416.Google Scholar
Degtyarëv, A. I., On deformations of singular plane sextics, J. Algebraic Geom. 17 (1) (2008), 101135.CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
The GAP Group, GAP – groups, algorithms, and programming, version 4.4, 2004, available at (http://www.gap-system.org).Google Scholar
Hirano, A., Construction of plane curves with cusps, Saitama Math. J. 10 (1992), 2124.Google Scholar
van Kampen, E. R., On the fundamental group of an algebraic curve, Amer. J. Math. 55 (1933), 255260.CrossRefGoogle Scholar
Krammer, D., The braid group ${B}_{4} $ is linear, Invent. Math. 142 (3) (2000), 451486.CrossRefGoogle Scholar
Kulikov, Vik. S., Generic coverings of the plane and braid monodromy invariants, in The Fano Conference, pp. 533558 (Univ. Torino, Turin, 2004).Google Scholar
Kulikov, Vik. S. and Teicher, M., Braid monodromy factorizations and diffeomorphism types, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2) (2000), 89120.Google Scholar
Lawrence, R. J., Homological representations of the Hecke algebra, Comm. Math. Phys. 135 (1) (1990), 141191.CrossRefGoogle Scholar
Libgober, A., On the Poincaré group of rational plane curves, Amer. J. Math. 58 (1936), 607619.Google Scholar
Libgober, A., The topological discriminant group of a Riemann surface of genus $p$ , Amer. J. Math. 59 (1937), 335358.Google Scholar
Libgober, A., On the homotopy type of the complement to plane algebraic curves, J. Reine Angew. Math. 367 (1986), 103114.Google Scholar
Libgober, A., Invariants of plane algebraic curves via representations of the braid groups, Invent. Math. 95 (1) (1989), 2530.CrossRefGoogle Scholar
Lönne, M., Fundamental groups of projective discriminant complements, Duke Math. J. 150 (2) (2009), 357405.CrossRefGoogle Scholar
MacLane, S., Some interpretations of abstract linear dependence in terms of projective geometry, Amer. J. Math. 58 (1) (1936), 236240.Google Scholar
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).CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
Rudolph, L., Algebraic functions and closed braids, Topology 22 (2) (1983), 191202.CrossRefGoogle Scholar
Salvetti, M., Arrangements of lines and monodromy of plane curves, Compositio Math. 68 (1) (1988), 103122.Google Scholar
Uludağ, A. M., More Zariski pairs and finite fundamental groups of curve complements, Manuscripta Math. 106 (3) (2001), 271277.CrossRefGoogle Scholar
Zariski, O., On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. Math. 51 (1929), 305328.CrossRefGoogle Scholar