Annales Mathématiques
Blaise Pascal
no. 1
Braid Monodromy of Algebraic Curves
José Ignacio Cogolludo-Agustín1
1 Departamento de Matemáticas, IUMA Universidad de Zaragoza C. Pedro Cerbuna, 12 50009 Zaragoza, Spain
Annales mathématiques Blaise Pascal, Tome 18 (2011) no. 1, pp. 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 : 10.5802/ambp.295
Classification : 32S50, 14D05, 14H30, 14H50, 32S05, 57M10
Mots clés : Fundamental group, algebraic variety, quasi-projective group, pencil of hypersurfaces

José Ignacio Cogolludo-Agustín 1

1 Departamento de Matemáticas, IUMA Universidad de Zaragoza C. Pedro Cerbuna, 12 50009 Zaragoza, Spain 
@article{AMBP_2011__18_1_141_0,
     author = {Jos\'e Ignacio Cogolludo-Agust{\'\i}n},
     title = {Braid {Monodromy} of {Algebraic} {Curves}},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {141--209},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {18},
     number = {1},
     year = {2011},
     doi = {10.5802/ambp.295},
     mrnumber = {2830090},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.295/}
}
TY  - JOUR
AU  - José Ignacio Cogolludo-Agustín
TI  - Braid Monodromy of Algebraic Curves
JO  - Annales mathématiques Blaise Pascal
PY  - 2011
SP  - 141
EP  - 209
VL  - 18
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.295/
DO  - 10.5802/ambp.295
LA  - en
ID  - AMBP_2011__18_1_141_0
ER  - 
%0 Journal Article
%A José Ignacio Cogolludo-Agustín
%T Braid Monodromy of Algebraic Curves
%J Annales mathématiques Blaise Pascal
%D 2011
%P 141-209
%V 18
%N 1
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.295/
%R 10.5802/ambp.295
%G en
%F AMBP_2011__18_1_141_0
José Ignacio Cogolludo-Agustín. Braid Monodromy of Algebraic Curves. Annales mathématiques Blaise Pascal, Tome 18 (2011) no. 1, pp. 141-209. doi : 10.5802/ambp.295. https://ambp.centre-mersenne.org/articles/10.5802/ambp.295/

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

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

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

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

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

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

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

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

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

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

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

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

[13] Ronald Brown 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 | Zbl

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

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

[16] D. Cheniot 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., Volume 27 (1973), pp. 141-158 | EuDML | Numdam | MR | Zbl

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

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

[19] Oscar Chisini 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), Volume 11(80) (1947), p. 3-6 (1949) | MR | Zbl

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

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

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

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

[24] Pierre Deligne 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 (Lecture Notes in Math.), Volume 842, Springer, Berlin, 1981, pp. 1-10 | EuDML | Numdam | MR | Zbl

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

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

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

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

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

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

[31] Michael Falk Homotopy types of line arrangements, Invent. Math., Volume 111 (1993) no. 1, pp. 139-150 | DOI | EuDML | MR | Zbl

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

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

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

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

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

[37] A Grothendieck; M. Raynaud Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3, Société Mathématique de France, Paris, 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 | Zbl

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

[39] Joe Harris On the Severi problem, Invent. Math., Volume 84 (1986) no. 3, pp. 445-461 | DOI | EuDML | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[66] Richard Randell 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., Volume 80 (1985) no. 3, pp. 467-468 | DOI | EuDML | MR | Zbl

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

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

[69] Mario Salvetti On the homotopy type of the complement to an arrangement of lines in C 2 , Boll. Un. Mat. Ital. A (7), Volume 2 (1988) no. 3, pp. 337-344 | MR | Zbl

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

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

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

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

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

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

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

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

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

Cité par Sources :