Braid Monodromy of Algebraic Curves
Annales mathématiques Blaise Pascal, Volume 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
Keywords: Fundamental group, algebraic variety, quasi-projective group, pencil of hypersurfaces
Cogolludo-Agustín, José Ignacio 1

1 Departamento de Matemáticas, IUMA Universidad de Zaragoza C. Pedro Cerbuna, 12 50009 Zaragoza, Spain
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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/articles/10.5802/ambp.295/

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

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

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

[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 (Adv. Stud. Pure Math.), Volume 43, Math. Soc. Japan, Tokyo, 2006, pp. 1-34 | MR

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

[6] Artal Bartolo, Enrique; Cogolludo, José Ignacio; Tokunaga, Hiro-o 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] Artin, E. Theory of braids, Ann. of Math. (2), Volume 48 (1947), pp. 101-126 | DOI | MR | Zbl

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

[9] Arvola, William A. 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] Ben-Itzhak, T.; Teicher, M. 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] Bessis, David Variations on Van Kampen’s method, J. Math. Sci. (N. Y.), Volume 128 (2005) no. 4, pp. 3142-3150 (Geometry) | DOI | MR | Zbl

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

[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 | Zbl

[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., Volume 53 (1986) no. 1, pp. 33-42 | DOI | MR | Zbl

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

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

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

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

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

[22] Cordovil, Raul 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] Coxeter, H. S. M.; Moser, W. O. J. 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] 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 (Lecture Notes in Math.), Volume 842, Springer, Berlin, 1981, pp. 1-10 | EuDML | Numdam | MR | Zbl

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

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

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

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

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

[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 | DOI | MR | Zbl

[37] Grothendieck, A; Raynaud, M. 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] Hamm, Helmut A. 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] Harris, Joe On the Severi problem, Invent. Math., Volume 84 (1986) no. 3, pp. 445-461 | DOI | EuDML | MR | Zbl

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

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

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

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

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

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

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

[50] Libgober, A. 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] Libgober, A. 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] Libgober, Anatoly 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] MacLane, Saunders 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] Manfredini, Sandro; Pignatelli, Roberto 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] Milnor, John Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J., 1968 | MR | Zbl

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

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

[60] Nemirovskiĭ, S. Yu. 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] Nori, Madhav V. 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] Orevkov, S. Yu. 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] Ran, Ziv 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] Randell, Richard 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] Randell, Richard 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] 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., Volume 80 (1985) no. 3, pp. 467-468 | DOI | EuDML | MR | Zbl

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

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

[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), Volume 2 (1988) no. 3, pp. 337-344 | MR | Zbl

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

[72] Severi, Francesco 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] 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) (Israel Math. Conf. Proc.), Volume 9, Bar-Ilan Univ., Ramat Gan (1996), pp. 393-416 | MR | Zbl

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

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

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