Braid monodromy of algebraic curves. (English) Zbl 1254.32043

Cogolludo-Agustín, José Ignacio

Ann. Math. Blaise Pascal 18, No. 1, 141-209 (2011).

The author gives a rather complete introductory course on the braid monodromy of algebraic curves.
The first section is a review of the basic tools necessary for the understanding of the subject: the fundamental groupoid and the Seifert-van Kampen theorem to compute it on the one side and branched and unbranched coverings of algebraic varieties on the other side. The problem of existence of maximal Galois branched coverings is given some attention. Eventually the monodromy action on a locally trivial fibration is introduced and, in the special case where the fibre is a disc without a finite set of points, it is interpreted as the action of the braid group.
The previous tools are used to construct from \(\mathbb{P}^2 \setminus \mathcal{C}\), where \(\mathcal{C}\) is an algebraic curve, a locally trivial fibration on \(\mathbb{P}^1\) minus a finite set of points and to prove the Zariski-van Kampen theorem, which allows to compute \(\pi_1(\mathbb{P}^2 \setminus \mathcal{C})\). Many examples of applications are given.
The third section contains further topics of the theory of braid monodromy. Notably a discussion of local and global monodromy, Hurwitz classes and wiring diagrams.
The whole article is completed with many figures that help to understand every concept explained.

Reviewer: Piotr P. Karwasz (Gdańsk)

MSC:

32S50	Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
14H30	Coverings of curves, fundamental group
14H50	Plane and space curves
14D05	Structure of families (Picard-Lefschetz, monodromy, etc.)
57M10	Covering spaces and low-dimensional topology
32S05	Local complex singularities

Keywords:

algebraic curve; fundamental group; monodromy; braid group

References:

[1]	Abelson, Harold, Topologically distinct conjugate varieties with finite fundamental group, Topology, 13, 161-176, (1974) · Zbl 0279.14001
[2]	Artal Bartolo, Enrique; Carmona Ruber, Jorge; Cogolludo Agustín, José Ignacio, Braid monodromy and topology of plane curves, Duke Math. J., 118, 2, 261-278, (2003) · 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., 141, 6, 1578-1588, (2005) · Zbl 1085.32012
[4]	Artal Bartolo, Enrique; Carmona Ruber, Jorge; Cogolludo Agustín, José Ignacio; Marco Buzunáriz, Miguel Ángel, Singularity theory and its applications, 43, Invariants of combinatorial line arrangements and rybnikov’s example, 1-34, (2006), Math. Soc. Japan, Tokyo · Zbl 1135.32025
[5]	Artal Bartolo, Enrique; Cogolludo, José Ignacio; Tokunaga, Hiro-o, Nodal degenerations of plane curves and Galois covers, Geom. Dedicata, 121, 129-142, (2006) · Zbl 1103.14016
[6]	Artal Bartolo, Enrique; Cogolludo, José Ignacio; Tokunaga, Hiro-o, Algebraic geometry in East Asia—Hanoi 2005, 50, A survey on Zariski pairs, 1-100, (2008), Math. Soc. Japan, Tokyo · Zbl 1141.14015
[7]	Artin, E., Theory of braids, Ann. of Math. (2), 48, 101-126, (1947) · Zbl 0030.17703
[8]	Arvola, William A., Complexified real arrangements of hyperplanes, Manuscripta Math., 71, 3, 295-306, (1991) · Zbl 0731.57011
[9]	Arvola, William A., The fundamental group of the complement of an arrangement of complex hyperplanes, Topology, 31, 4, 757-765, (1992) · Zbl 0772.57001
[10]	Ben-Itzhak, T.; Teicher, M., Properties of Hurwitz equivalence in the braid group of order \(n,\) J. Algebra, 264, 1, 15-25, (2003) · Zbl 1054.20017
[11]	Bessis, David, Variations on Van kampen’s method, J. Math. Sci. (N. Y.), 128, 4, 3142-3150, (2005) · Zbl 1121.57002
[12]	Birman, Joan S., Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math., 22, 213-238, (1969) · Zbl 0167.21503
[13]	Brown, Ronald, Topology and groupoids, (2006), BookSurge, LLC, Charleston, SC · Zbl 1093.55001
[14]	Carmona Ruber, J., Monodromía de trenzas de curvas algebraicas planas, (2003)
[15]	Catanese, F., On a problem of chisini, Duke Math. J., 53, 1, 33-42, (1986) · 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., 27, 141-158, (1973) · Zbl 0294.14010
[17]	Chéniot, D.; Libgober, A., Zariski-Van kampen theorem for higher-homotopy groups, J. Inst. Math. Jussieu, 2, 4, 495-527, (2003) · Zbl 1081.14505
[18]	Chisini, Oscar, Una suggestiva rappresentazione reale per le curve algebriche piane, Ist. Lombardo, Rend., II. Ser., 66), 1141-1155, (1933) · 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), 11(80), 3-6 (1949), (1947) · Zbl 0041.28002
[20]	Cohen, Daniel C.; Suciu, Alexander I., The braid monodromy of plane algebraic curves and hyperplane arrangements, Comment. Math. Helv., 72, 2, 285-315, (1997) · Zbl 0959.52018
[21]	Cordovil, R.; Fachada, J. L., Braid monodromy groups of wiring diagrams, Boll. Un. Mat. Ital. B (7), 9, 2, 399-416, (1995) · 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., 21, 3, 481-498, (1998) · Zbl 0921.55004
[23]	Coxeter, H. S. M.; Moser, W. O. J., Generators and relations for discrete groups, 14, (1980), Springer-Verlag, Berlin · Zbl 0422.20001
[24]	Deligne, Pierre, Bourbaki Seminar, Vol. 1979/80, 842, Le groupe fondamental du complément d’une courbe plane n’ayant que des points doubles ordinaires est abélien (d’après W. fulton), 1-10, (1981), Springer, Berlin · Zbl 0478.14008
[25]	Dimca, Alexandru, Singularities and topology of hypersurfaces, (1992), Springer-Verlag, New York · Zbl 0753.57001
[26]	Dolgachev, Igor; Libgober, Anatoly, Algebraic geometry (Chicago, Ill., 1980), 862, On the fundamental group of the complement to a discriminant variety, 1-25, (1981), Springer, Berlin · Zbl 0475.14011
[27]	Dunwoody, M. J., The homotopy type of a two-dimensional complex, Bull. London Math. Soc., 8, 3, 282-285, (1976) · Zbl 0341.55008
[28]	Ehresmann, Charles, Sur LES espaces fibrés différentiables, C. R. Acad. Sci. Paris, 224, 1611-1612, (1947) · Zbl 0029.42001
[29]	Enriques, Federigo, Sulla costruzione delle funzioni algebriche di due variabili possedenti una data curva di diramazione, Ann. Mat. Pura Appl., 1, 1, 185-198, (1924) · JFM 50.0674.01
[30]	Falk, Michael, The minimal model of the complement of an arrangement of hyperplanes, Trans. Amer. Math. Soc., 309, 2, 543-556, (1988) · Zbl 0707.57001
[31]	Falk, Michael, Homotopy types of line arrangements, Invent. Math., 111, 1, 139-150, (1993) · Zbl 0772.52011
[32]	Fulton, William, On the fundamental group of the complement of a node curve, Ann. of Math. (2), 111, 2, 407-409, (1980) · Zbl 0406.14008
[33]	Goresky, Mark; MacPherson, Robert, Stratified Morse theory, 14, (1988), Springer-Verlag, Berlin · 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., 350, 1, 251-274, (1998) · Zbl 0889.14010
[35]	Greuel, Gert-Martin; Lossen, Christoph; Shustin, Eugenii, Plane curves of minimal degree with prescribed singularities, Invent. Math., 133, 3, 539-580, (1998) · 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, 11, 543-550, (2001) · Zbl 0982.14018
[37]	Grothendieck, A; Raynaud, M., Revêtements étales et groupe fondamental (SGA 1), (2003), Société Mathématique de France, Paris · Zbl 0234.14002
[38]	Hamm, Helmut A., Singularities, Part 1 (Arcata, Calif., 1981), 40, Lefschetz theorems for singular varieties, 547-557, (1983), Amer. Math. Soc., Providence, RI · Zbl 0525.14011
[39]	Harris, Joe, On the Severi problem, Invent. Math., 84, 3, 445-461, (1986) · Zbl 0596.14017
[40]	Hironaka, Eriko, Abelian coverings of the complex projective plane branched along configurations of real lines, Mem. Amer. Math. Soc., 105, 502, vi+85 pp., (1993) · Zbl 0788.14054
[41]	van Kampen, Egbert R., On the connection between the fundamental groups of some related spaces., Am. J. Math., 55, 261-267, (1933) · Zbl 0006.41503
[42]	Kampen, Egbert R. Van, On the fundamental group of an algebraic curve, Amer. J. Math., 55, 1-4, 255-260, (1933) · 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., 333, 9, 855-859, (2001) · Zbl 1066.14050
[44]	Kulikov, Valentine S., Real and complex singularities, 232, On a conjecture of chisini for coverings of the plane with A-D-E-singularities, 175-188, (2003), Dekker, New York · Zbl 1081.14050
[45]	Kulikov, Vik. S., On chisini’s conjecture, Izv. Ross. Akad. Nauk Ser. Mat., 63, 6, 83-116, (1999) · Zbl 0962.14005
[46]	Kulikov, Vik. S., On chisini’s conjecture. II, Izv. Ross. Akad. Nauk Ser. Mat., 72, 5, 63-76, (2008) · Zbl 1153.14012
[47]	Kulikov, Vik. S.; Kharlamov, V. M., On braid monodromy factorizations, Izv. Ross. Akad. Nauk Ser. Mat., 67, 3, 79-118, (2003) · Zbl 1076.14022
[48]	Kulikov, Vik. S.; Taĭkher, M., Braid monodromy factorizations and diffeomorphism types, Izv. Ross. Akad. Nauk Ser. Mat., 64, 2, 89-120, (2000) · Zbl 1004.14005
[49]	Lamotke, Klaus, The topology of complex projective varieties after S. Lefschetz, Topology, 20, 1, 15-51, (1981) · Zbl 0445.14010
[50]	Libgober, A., On the homotopy type of the complement to plane algebraic curves, J. Reine Angew. Math., 367, 103-114, (1986) · Zbl 0576.14019
[51]	Libgober, A., Homotopy groups of the complements to singular hypersurfaces. II, Ann. of Math. (2), 139, 1, 117-144, (1994) · Zbl 0815.57017
[52]	Libgober, Anatoly, Singularity theory and its applications, 43, Homotopy groups of complements to ample divisors, 179-204, (2006), Math. Soc. Japan, Tokyo · Zbl 1134.14014
[53]	MacLane, Saunders, Some interpretations of abstract linear dependence in terms of projective geometry, Amer. J. Math., 58, 1, 236-240, (1936) · Zbl 0013.19503
[54]	Manfredini, Sandro; Pignatelli, Roberto, Chisini’s conjecture for curves with singularities of type \(x^n=y^m,\) Michigan Math. J., 50, 2, 287-312, (2002) · Zbl 1065.14045
[55]	Milnor, John, Singular points of complex hypersurfaces, (1968), Princeton University Press, Princeton, N.J. · Zbl 0184.48405
[56]	Moishezon, B., Geometric topology (Haifa, 1992), 164, The arithmetic of braids and a statement of chisini, 151-175, (1994), Amer. Math. Soc., Providence, RI · Zbl 0837.14020
[57]	Moishezon, B. G., Algebraic geometry (Chicago, Ill., 1980), 862, Stable branch curves and braid monodromies, 107-192, (1981), Springer, Berlin · Zbl 0476.14005
[58]	Munkres, James R., Topology: a first course, (1975), Prentice-Hall Inc., Englewood Cliffs, N.J. · Zbl 0306.54001
[59]	Namba, Makoto, Branched coverings and algebraic functions, 161, (1987), Longman Scientific & Technical, Harlow · Zbl 0706.14017
[60]	Nemirovskiĭ, S. Yu., On kulikov’s theorem on the chisini conjecture, Izv. Ross. Akad. Nauk Ser. Mat., 65, 1, 77-80, (2001) · Zbl 1012.14005
[61]	Nori, Madhav V., Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. (4), 16, 2, 305-344, (1983) · 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., 326, 7, 867-871, (1998) · Zbl 0922.32020
[63]	Ran, Ziv, Families of plane curves and their limits: enriques’ conjecture and beyond, Ann. of Math. (2), 130, 1, 121-157, (1989) · Zbl 0704.14018
[64]	Randell, Richard, The fundamental group of the complement of a union of complex hyperplanes, Invent. Math., 69, 1, 103-108, (1982) · Zbl 0505.14017
[65]	Randell, Richard, Singularities, Part 2 (Arcata, Calif., 1981), 40, Milnor fibers and Alexander polynomials of plane curves, 415-419, (1983), Amer. Math. Soc., Providence, RI · 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., 80, 3, 467-468, (1985) · Zbl 0596.14014
[67]	Rybnikov, G., On the fundamental group of the complement of a complex hyperplane arrangement · Zbl 1271.14085
[68]	Salvetti, Mario, Arrangements of lines and monodromy of plane curves, Compositio Math., 68, 1, 103-122, (1988) · Zbl 0661.14038
[69]	Salvetti, Mario, On the homotopy type of the complement to an arrangement of lines in \(\textbf{C}^2,\) Boll. Un. Mat. Ital. A (7), 2, 3, 337-344, (1988) · Zbl 0668.57002
[70]	Seifert, H., Konstruktion dreidimensionaler geschlossener Räume, Berichte über d. Verhandl. d. Sächs. Ges. d. Wiss., Math.-Phys. Kl., 83, 26-66, (1931) · JFM 57.0723.01
[71]	Serre, Jean-Pierre, Exemples de variétés projectives conjuguées non homéomorphes, C. R. Acad. Sci. Paris, 258, 4194-4196, (1964) · Zbl 0117.38003
[72]	Severi, Francesco, Vorlesungen über algebraische Geometrie: Geometrie auf einer Kurve, Riemannsche Flächen, Abelsche Integrale, (1968), Johnson Reprint Corp., New York
[73]	Shimada, I., Lecture on Zariski Van-Kampen theorem, (2007)
[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), 9, 393-416, (1996), Bar-Ilan Univ., Ramat Gan · Zbl 0857.14015
[75]	Vassiliev, V. A., Introduction to topology, 14, (2001), American Mathematical Society, Providence, RI · 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., 51, 2, 305-328, (1929) · JFM 55.0806.01
[77]	Zariski, Oscar, On the irregularity of cyclic multiple planes, Ann. of Math. (2), 32, 3, 485-511, (1931) · Zbl 0001.40301
[78]	Zariski, Oscar, On the Poincaré group of rational plane curves, Amer. J. Math., 58, 3, 607-619, (1936) · Zbl 0014.32801