Braid Monodromy of Algebraic Curves
Annales mathématiques Blaise Pascal, Volume 18 (2011) no. 1, p. 141-209

These are the notes from a one-week course on Braid Monodromy of Algebraic Curves given at the Université de Pau et des Pays de l’Adour during the Première Ecole Franco-Espagnole: Groupes de tresses et topologie en petite dimension in October 2009.

This is intended to be an introductory survey through which we hope we can briefly outline the power of the concept monodromy as a common area for group theory, algebraic geometry, and topology of projective curves.

The main classical results are stated in §2, where the Zariski–van Kampen method to compute a presentation for the fundamental group of the complement to projective plane curves is presented. In §1 these results are prefaced with a review of basic concepts like fundamental groups, locally trivial fibrations, branched and unbranched coverings and a first peek at monodromy. Descriptions of the main motivations that have lead mathematicians to study these objects are included throughout this first chapter. Finally, additional tools and further results that are direct applications of braid monodromy will be considered in §3.

While not all proofs are included, we do provide either originals or simplified versions of those that are relevant in the sense that they exhibit the techniques that are most used in this context and lead to a better understanding of the main concepts discussed in this survey.

Nothing here is hence original, other than an attempt to bring together different results and points of view.

It goes without saying that this is not the first, and hopefully not the last, survey on the topic. For other approaches to braid monodromy we refer to the following beautifully-written papers [73, 20, 6].

We finally wish to thank the organizers and the referee for their patience and understanding in the process of writing and correcting these notes.

DOI : https://doi.org/10.5802/ambp.295
Classification:  32S50,  14D05,  14H30,  14H50,  32S05,  57M10
Keywords: Fundamental group, algebraic variety, quasi-projective group, pencil of hypersurfaces
@article{AMBP_2011__18_1_141_0,
author = {Cogolludo-Agust\'\i n, Jos\'e Ignacio},
title = {Braid Monodromy of Algebraic Curves},
journal = {Annales math\'ematiques Blaise Pascal},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {18},
number = {1},
year = {2011},
pages = {141-209},
doi = {10.5802/ambp.295},
mrnumber = {2830090},
zbl = {pre05903955},
language = {en},
url = {http://www.numdam.org/item/AMBP_2011__18_1_141_0}
}

Cogolludo-Agustín, José Ignacio. Braid Monodromy of Algebraic Curves. Annales mathématiques Blaise Pascal, Volume 18 (2011) no. 1, pp. 141-209. doi : 10.5802/ambp.295. http://www.numdam.org/item/AMBP_2011__18_1_141_0/

[1] Abelson, Harold Topologically distinct conjugate varieties with finite fundamental group, Topology, Tome 13 (1974), pp. 161-176 | Article | MR 349679 | Zbl 0279.14001

[2] Artal Bartolo, Enrique; Carmona Ruber, Jorge; Cogolludo Agustín, José Ignacio Braid monodromy and topology of plane curves, Duke Math. J., Tome 118 (2003) no. 2, pp. 261-278 | Article | MR 1980995 | Zbl 1058.14053

[3] Artal Bartolo, Enrique; Carmona Ruber, Jorge; Cogolludo-Agustín, José Ignacio; Marco Buzunáriz, Miguel Topology and combinatorics of real line arrangements, Compos. Math., Tome 141 (2005) no. 6, pp. 1578-1588 | Article | MR 2188450 | Zbl 1085.32012

[4] Artal Bartolo, Enrique; Carmona Ruber, Jorge; Cogolludo Agustín, José Ignacio; Marco Buzunáriz, Miguel Ángel Invariants of combinatorial line arrangements and Rybnikov’s example, Singularity theory and its applications, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 43 (2006), pp. 1-34 | MR 2313406

[5] Artal Bartolo, Enrique; Cogolludo, José Ignacio; Tokunaga, Hiro-O Nodal degenerations of plane curves and Galois covers, Geom. Dedicata, Tome 121 (2006), pp. 129-142 | Article | MR 2276239 | Zbl 1103.14016

[6] Artal Bartolo, Enrique; Cogolludo, José Ignacio; Tokunaga, Hiro-O A survey on Zariski pairs, Algebraic geometry in East Asia—Hanoi 2005, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 50 (2008), pp. 1-100 | MR 2409555

[7] Artin, E. Theory of braids, Ann. of Math. (2), Tome 48 (1947), pp. 101-126 | Article | MR 19087 | Zbl 0030.17703

[8] Arvola, William A. Complexified real arrangements of hyperplanes, Manuscripta Math., Tome 71 (1991) no. 3, pp. 295-306 | Article | MR 1103735 | Zbl 0731.57011

[9] Arvola, William A. The fundamental group of the complement of an arrangement of complex hyperplanes, Topology, Tome 31 (1992) no. 4, pp. 757-765 | Article | MR 1191377 | Zbl 0772.57001

[10] Ben-Itzhak, T.; Teicher, M. Properties of Hurwitz equivalence in the braid group of order $n$, J. Algebra, Tome 264 (2003) no. 1, pp. 15-25 | Article | MR 1980683 | Zbl 1054.20017

[11] Bessis, David Variations on Van Kampen’s method, J. Math. Sci. (N. Y.), Tome 128 (2005) no. 4, pp. 3142-3150 (Geometry) | Article | MR 2171593 | Zbl 1121.57002

[12] Birman, Joan S. Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math., Tome 22 (1969), pp. 213-238 | Article | MR 243519 | Zbl 0167.21503

[13] Brown, Ronald Topology and groupoids, BookSurge, LLC, Charleston, SC (2006) (Third edition of ıt Elements of modern topology [McGraw-Hill, New York, 1968; MR0227979], With 1 CD-ROM (Windows, Macintosh and UNIX)) | MR 2273730 | Zbl 1093.55001

[14] Carmona Ruber, J. Monodromía de trenzas de curvas algebraicas planas, Universidad de Zaragoza (2003) (Ph. D. Thesis)

[15] Catanese, F. On a problem of Chisini, Duke Math. J., Tome 53 (1986) no. 1, pp. 33-42 | Article | MR 835794 | Zbl 0609.14031

[16] Cheniot, D. Une démonstration du théorème de Zariski sur les sections hyperplanes d’une hypersurface projective et du théorème de Van Kampen sur le groupe fondamental du complémentaire d’une courbe projective plane, Compositio Math., Tome 27 (1973), pp. 141-158 | Numdam | MR 366922 | Zbl 0294.14010

[17] Chéniot, D.; Libgober, A. Zariski-van Kampen theorem for higher-homotopy groups, J. Inst. Math. Jussieu, Tome 2 (2003) no. 4, pp. 495-527 | Article | MR 2006797 | Zbl 1081.14505

[18] Chisini, Oscar Una suggestiva rappresentazione reale per le curve algebriche piane, Ist. Lombardo, Rend., II. Ser., Tome 66) (1933), pp. 1141-1155 | Zbl 0008.22001

[19] Chisini, Oscar Sulla identità birazionale di due funzioni algebriche di più variabili, dotate di una medesima varietà di diramazione, Ist. Lombardo Sci. Lett. Rend Cl. Sci. Mat. Nat. (3), Tome 11(80) (1947), p. 3-6 (1949) | MR 34054 | Zbl 0041.28002

[20] Cohen, Daniel C.; Suciu, Alexander I. The braid monodromy of plane algebraic curves and hyperplane arrangements, Comment. Math. Helv., Tome 72 (1997) no. 2, pp. 285-315 | Article | MR 1470093 | Zbl 0959.52018

[21] Cordovil, R.; Fachada, J. L. Braid monodromy groups of wiring diagrams, Boll. Un. Mat. Ital. B (7), Tome 9 (1995) no. 2, pp. 399-416 | MR 1333969 | Zbl 0868.14028

[22] Cordovil, Raul The fundamental group of the complement of the complexification of a real arrangement of hyperplanes, Adv. in Appl. Math., Tome 21 (1998) no. 3, pp. 481-498 | Article | MR 1641238 | Zbl 0921.55004

[23] Coxeter, H. S. M.; Moser, W. O. J. Generators and relations for discrete groups, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Tome 14 (1980) | MR 562913 | Zbl 0422.20001 | Zbl 0077.02801

[24] Deligne, Pierre Le groupe fondamental du complément d’une courbe plane n’ayant que des points doubles ordinaires est abélien (d’après W. Fulton), Bourbaki Seminar, Vol. 1979/80, Springer, Berlin (Lecture Notes in Math.) Tome 842 (1981), pp. 1-10 | Numdam | MR 636513 | Zbl 0478.14008

[25] Dimca, Alexandru Singularities and topology of hypersurfaces, Springer-Verlag, New York, Universitext (1992) | MR 1194180 | Zbl 0753.57001

[26] Dolgachev, Igor; Libgober, Anatoly On the fundamental group of the complement to a discriminant variety, Algebraic geometry (Chicago, Ill., 1980), Springer, Berlin (Lecture Notes in Math.) Tome 862 (1981), pp. 1-25 | MR 644816 | Zbl 0475.14011

[27] Dunwoody, M. J. The homotopy type of a two-dimensional complex, Bull. London Math. Soc., Tome 8 (1976) no. 3, pp. 282-285 | Article | MR 425943 | Zbl 0341.55008

[28] Ehresmann, Charles Sur les espaces fibrés différentiables, C. R. Acad. Sci. Paris, Tome 224 (1947), p. 1611-1612 | MR 20774 | Zbl 0029.42001

[29] Enriques, Federigo Sulla costruzione delle funzioni algebriche di due variabili possedenti una data curva di diramazione, Ann. Mat. Pura Appl., Tome 1 (1924) no. 1, pp. 185-198 | Article | JFM 50.0674.01 | MR 1553061

[30] Falk, Michael The minimal model of the complement of an arrangement of hyperplanes, Trans. Amer. Math. Soc., Tome 309 (1988) no. 2, pp. 543-556 | Article | MR 929668 | Zbl 0707.57001

[31] Falk, Michael Homotopy types of line arrangements, Invent. Math., Tome 111 (1993) no. 1, pp. 139-150 | Article | MR 1193601 | Zbl 0772.52011

[32] Fulton, William On the fundamental group of the complement of a node curve, Ann. of Math. (2), Tome 111 (1980) no. 2, pp. 407-409 | Article | MR 569076 | Zbl 0406.14008

[33] Goresky, Mark; Macpherson, Robert Stratified Morse theory, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 14 (1988) | MR 932724 | Zbl 0639.14012

[34] Greuel, Gert-Martin; Lossen, Christoph; Shustin, Eugenii Geometry of families of nodal curves on the blown-up projective plane, Trans. Amer. Math. Soc., Tome 350 (1998) no. 1, pp. 251-274 | Article | MR 1443875 | Zbl 0889.14010

[35] Greuel, Gert-Martin; Lossen, Christoph; Shustin, Eugenii Plane curves of minimal degree with prescribed singularities, Invent. Math., Tome 133 (1998) no. 3, pp. 539-580 | Article | MR 1645074 | Zbl 0924.14013

[36] Greuel, Gert-Martin; Lossen, Christoph; Shustin, Eugenii The variety of plane curves with ordinary singularities is not irreducible, Internat. Math. Res. Notices (2001) no. 11, pp. 543-550 | Article | MR 1836729 | Zbl 0982.14018

[37] Grothendieck, A; Raynaud, M. Revêtements étales et groupe fondamental (SGA 1), Société Mathématique de France, Paris, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3 (2003) (Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]) | MR 2017446 | Zbl 0234.14002

[38] Hamm, Helmut A. Lefschetz theorems for singular varieties, Singularities, Part 1 (Arcata, Calif., 1981), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 40 (1983), pp. 547-557 | MR 713091 | Zbl 0525.14011

[39] Harris, Joe On the Severi problem, Invent. Math., Tome 84 (1986) no. 3, pp. 445-461 | Article | MR 837522 | Zbl 0596.14017

[40] Hironaka, Eriko Abelian coverings of the complex projective plane branched along configurations of real lines, Mem. Amer. Math. Soc., Tome 105 (1993) no. 502, pp. vi+85 | MR 1164128 | Zbl 0788.14054

[41] Van Kampen, Egbert R. On the connection between the fundamental groups of some related spaces., Am. J. Math., Tome 55 (1933), pp. 261-267 | Zbl 0006.41503

[42] Kampen, Egbert R. Van On the Fundamental Group of an Algebraic Curve, Amer. J. Math., Tome 55 (1933) no. 1-4, pp. 255-260 | Article | MR 1506962 | Zbl 0006.41502

[43] Kharlamov, Viatcheslav; Kulikov, Viktor Diffeomorphisms, isotopies, and braid monodromy factorizations of plane cuspidal curves, C. R. Acad. Sci. Paris Sér. I Math., Tome 333 (2001) no. 9, pp. 855-859 | Article | MR 1873224 | Zbl 1066.14050

[44] Kulikov, Valentine S. On a conjecture of Chisini for coverings of the plane with A-D-E-singularities, Real and complex singularities, Dekker, New York (Lecture Notes in Pure and Appl. Math.) Tome 232 (2003), pp. 175-188 | MR 2075064 | Zbl 1081.14050

[45] Kulikov, Vik. S. On Chisini’s conjecture, Izv. Ross. Akad. Nauk Ser. Mat., Tome 63 (1999) no. 6, pp. 83-116 | Article | MR 1748562 | Zbl 0962.14005

[46] Kulikov, Vik. S. On Chisini’s conjecture. II, Izv. Ross. Akad. Nauk Ser. Mat., Tome 72 (2008) no. 5, pp. 63-76 | Article | MR 2473772 | Zbl 1153.14012

[47] Kulikov, Vik. S.; Kharlamov, V. M. On braid monodromy factorizations, Izv. Ross. Akad. Nauk Ser. Mat., Tome 67 (2003) no. 3, pp. 79-118 | Article | MR 1992194 | Zbl 1076.14022

[48] Kulikov, Vik. S.; Taĭkher, M. Braid monodromy factorizations and diffeomorphism types, Izv. Ross. Akad. Nauk Ser. Mat., Tome 64 (2000) no. 2, pp. 89-120 | Article | MR 1770673 | Zbl 1004.14005

[49] Lamotke, Klaus The topology of complex projective varieties after S. Lefschetz, Topology, Tome 20 (1981) no. 1, pp. 15-51 | Article | MR 592569 | Zbl 0445.14010

[50] Libgober, A. On the homotopy type of the complement to plane algebraic curves, J. Reine Angew. Math., Tome 367 (1986), pp. 103-114 | Article | MR 839126 | Zbl 0576.14019

[51] Libgober, A. Homotopy groups of the complements to singular hypersurfaces. II, Ann. of Math. (2), Tome 139 (1994) no. 1, pp. 117-144 | Article | MR 1259366 | Zbl 0815.57017

[52] Libgober, Anatoly Homotopy groups of complements to ample divisors, Singularity theory and its applications, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 43 (2006), pp. 179-204 | MR 2325138 | Zbl 1134.14014

[53] Maclane, Saunders Some Interpretations of Abstract Linear Dependence in Terms of Projective Geometry, Amer. J. Math., Tome 58 (1936) no. 1, pp. 236-240 | Article | MR 1507146 | Zbl 0013.19503

[54] Manfredini, Sandro; Pignatelli, Roberto Chisini’s conjecture for curves with singularities of type ${x}^{n}={y}^{m}$, Michigan Math. J., Tome 50 (2002) no. 2, pp. 287-312 | Article | MR 1914066 | Zbl 1065.14045

[55] Milnor, John Singular points of complex hypersurfaces, Princeton University Press, Princeton, N.J., Annals of Mathematics Studies, No. 61 (1968) | MR 239612 | Zbl 0184.48405

[56] Moishezon, B. The arithmetic of braids and a statement of Chisini, Geometric topology (Haifa, 1992), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 164 (1994), pp. 151-175 | MR 1282761 | Zbl 0837.14020

[57] Moishezon, B. G. Stable branch curves and braid monodromies, Algebraic geometry (Chicago, Ill., 1980), Springer, Berlin (Lecture Notes in Math.) Tome 862 (1981), pp. 107-192 | MR 644819 | Zbl 0476.14005

[58] Munkres, James R. Topology: a first course, Prentice-Hall Inc., Englewood Cliffs, N.J. (1975) | MR 464128 | Zbl 0306.54001 | Zbl 0951.54001

[59] Namba, Makoto Branched coverings and algebraic functions, Longman Scientific & Technical, Harlow, Pitman Research Notes in Mathematics Series, Tome 161 (1987) | MR 933557 | Zbl 0706.14017

[60] Nemirovskiĭ, S. Yu. On Kulikov’s theorem on the Chisini conjecture, Izv. Ross. Akad. Nauk Ser. Mat., Tome 65 (2001) no. 1, pp. 77-80 | Article | MR 1829404 | Zbl 1012.14005

[61] Nori, Madhav V. Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. (4), Tome 16 (1983) no. 2, pp. 305-344 | Numdam | MR 732347 | Zbl 0527.14016

[62] Orevkov, S. Yu. Realizability of a braid monodromy by an algebraic function in a disk, C. R. Acad. Sci. Paris Sér. I Math., Tome 326 (1998) no. 7, pp. 867-871 | Article | MR 1648548 | Zbl 0922.32020

[63] Ran, Ziv Families of plane curves and their limits: Enriques’ conjecture and beyond, Ann. of Math. (2), Tome 130 (1989) no. 1, pp. 121-157 | Article | MR 1005609 | Zbl 0704.14018

[64] Randell, Richard The fundamental group of the complement of a union of complex hyperplanes, Invent. Math., Tome 69 (1982) no. 1, pp. 103-108 | Article | MR 671654 | Zbl 0505.14017

[65] Randell, Richard Milnor fibers and Alexander polynomials of plane curves, Singularities, Part 2 (Arcata, Calif., 1981), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 40 (1983), pp. 415-419 | MR 713266 | Zbl 0524.14027

[66] Randell, Richard Correction: “The fundamental group of the complement of a union of complex hyperplanes” [Invent. Math. 69 (1982), no. 1, 103–108; MR0671654 (84a:32016)], Invent. Math., Tome 80 (1985) no. 3, p. 467-468 | Article | MR 671654 | Zbl 0596.14014

[67] Rybnikov, G. On the fundamental group of the complement of a complex hyperplane arrangement (Preprint available at arXiv:math.AG/9805056) | Zbl 1271.14085

[68] Salvetti, Mario Arrangements of lines and monodromy of plane curves, Compositio Math., Tome 68 (1988) no. 1, pp. 103-122 | Numdam | MR 962507 | Zbl 0661.14038

[69] Salvetti, Mario On the homotopy type of the complement to an arrangement of lines in ${\mathbf{C}}^{2}$, Boll. Un. Mat. Ital. A (7), Tome 2 (1988) no. 3, pp. 337-344 | MR 966915 | Zbl 0668.57002

[70] Seifert, H. Konstruktion dreidimensionaler geschlossener Räume, Berichte über d. Verhandl. d. Sächs. Ges. d. Wiss., Math.-Phys. Kl., Tome 83 (1931), pp. 26-66 | JFM 57.0723.01 | Zbl 0002.16001

[71] Serre, Jean-Pierre Exemples de variétés projectives conjuguées non homéomorphes, C. R. Acad. Sci. Paris, Tome 258 (1964), pp. 4194-4196 | MR 166197 | Zbl 0117.38003

[72] Severi, Francesco Vorlesungen über algebraische Geometrie: Geometrie auf einer Kurve, Riemannsche Flächen, Abelsche Integrale, Johnson Reprint Corp., New York, Berechtigte Deutsche Übersetzung von Eugen Löffler. Mit einem Einführungswort von A. Brill. Begleitwort zum Neudruck von Beniamino Segre. Bibliotheca Mathematica Teubneriana, Band 32 (1968) | MR 245574

[73] Shimada, I. Lecture on Zariski Van-Kampen theorem (2007) (Lectures Notes)

[74] Shustin, Eugenii Smoothness and irreducibility of families of plane algebraic curves with ordinary singularities, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Bar-Ilan Univ., Ramat Gan (Israel Math. Conf. Proc.) Tome 9 (1996), pp. 393-416 | MR 1360516 | Zbl 0857.14015

[75] Vassiliev, V. A. Introduction to topology, American Mathematical Society, Providence, RI, Student Mathematical Library, Tome 14 (2001) (Translated from the 1997 Russian original by A. Sossinski) | MR 1816237 | Zbl 0971.57001

[76] Zariski, Oscar On the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve, Amer. J. Math., Tome 51 (1929) no. 2, pp. 305-328 | Article | JFM 55.0806.01 | MR 1506719

[77] Zariski, Oscar On the irregularity of cyclic multiple planes, Ann. of Math. (2), Tome 32 (1931) no. 3, pp. 485-511 | Article | MR 1503012 | Zbl 0001.40301

[78] Zariski, Oscar On the Poincaré Group of Rational Plane Curves, Amer. J. Math., Tome 58 (1936) no. 3, pp. 607-619 | Article | MR 1507185 | Zbl 0014.32801