Book Review
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3196796
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Book Information:
Author:
Alex Degtyarev
Title:
Topology of algebraic curves. An approach via dessins d’enfants
Additional book information:
Studies in Mathematics, Vol. 44,
Walter de Gruyter \& Co.,
Berlin,
2012,
xvi+393 pp.,
ISBN 978-3-11-025591-1,
US $181.00
J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), no. 2, 275–306. MR 1501429, DOI 10.1090/S0002-9947-1928-1501429-1
Enrique Artal Bartolo and José Ignacio Cogolludo-Agustín, On the connection between fundamental groups and pencils with multiple fibers, J. Singul. 2 (2010), 1–18. MR 2763015, DOI 10.5427/jsing.2010.2a
Denis Auroux, Symplectic 4-manifolds as branched coverings of $\mathbf C\mathbf P^2$, Invent. Math. 139 (2000), no. 3, 551–602. MR 1738061, DOI 10.1007/s002220050019
D. Auroux, S. K. Donaldson, and L. Katzarkov, Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves, Math. Ann. 326 (2003), no. 1, 185–203. MR 1981618, DOI 10.1007/s00208-003-0418-9
E. Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math. 12 (1971), 57–61 (German). MR 293615, DOI 10.1007/BF01389827
Fabrizio Catanese, Differentiable and deformation type of algebraic surfaces, real and symplectic structures, Symplectic 4-manifolds and algebraic surfaces, Lecture Notes in Math., vol. 1938, Springer, Berlin, 2008, pp. 55–167. MR 2441412, DOI 10.1007/978-3-540-78279-7_{2}
Oscar Chisini, Il teorema di esistenza delle trecce algebriche. I, II, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 17 (1954), 143–149, 307–311 (1955) (Italian). MR 71103
J. I. Cogolludo-Agustin and A. Libgober, Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves, J. Reine Angew. Math., to appear.
P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240
Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
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., vol. 842, Springer, Berlin-New York, 1981, pp. 1–10 (French). MR 636513
Pierre Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273–302 (French). MR 422673, DOI 10.1007/BF01406236
Alexandru Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. MR 1194180, DOI 10.1007/978-1-4612-4404-2
Alexandru Dimca, Characteristic varieties and logarithmic differential 1-forms, Compos. Math. 146 (2010), no. 1, 129–144. MR 2581244, DOI 10.1112/S0010437X09004461
Alexandru Dimca and Alexander I. Suciu, Which 3-manifold groups are Kähler groups?, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 3, 521–528. MR 2505439, DOI 10.4171/JEMS/158
F. Enriques, Sulla construzione delle funzioni algebriche di due variabili possedenti una data curva diramazione, Ann. Nat. Pure Appl. vol. 1 (1923).
Robert Friedman and John W. Morgan, Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 27, Springer-Verlag, Berlin, 1994. MR 1288304, DOI 10.1007/978-3-662-03028-8
William Fulton, On the fundamental group of the complement of a node curve, Ann. of Math. (2) 111 (1980), no. 2, 407–409. MR 569076, DOI 10.2307/1971204
Leila Schneps and Pierre Lochak (eds.), Geometric Galois actions. 1, London Mathematical Society Lecture Note Series, vol. 242, Cambridge University Press, Cambridge, 1997. Around Grothendieck’s “Esquisse d’un programme”. MR 1483106, DOI 10.1017/CBO9780511758874
Joe Harris, On the Severi problem, Invent. Math. 84 (1986), no. 3, 445–461. MR 837522, DOI 10.1007/BF01388741
Eriko Hironaka, Alexander stratifications of character varieties, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 2, 555–583. MR 1450425
F. Hirzebruch, Arrangements of lines and algebraic surfaces, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser, Boston, Mass., 1983, pp. 113–140. MR 717609
Vik. S. Kulikov, On Chisini’s conjecture, Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), no. 6, 83–116 (Russian, with Russian summary); English transl., Izv. Math. 63 (1999), no. 6, 1139–1170. MR 1748562, DOI 10.1070/im1999v063n06ABEH000267
Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J. 49 (1982), no. 4, 833–851. MR 683005
A. Libgober, Alexander invariants of plane algebraic curves, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 135–143. MR 713242
A. Libgober, Characteristic varieties of algebraic curves, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001) NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 215–254. MR 1866902
Anatoly Libgober, Non vanishing loci of Hodge numbers of local systems, Manuscripta Math. 128 (2009), no. 1, 1–31. MR 2470184, DOI 10.1007/s00229-008-0221-8
Anatoly Libgober and Sergey Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121 (2000), no. 3, 337–361. MR 1761630, DOI 10.1023/A:1001826010964
B. G. Moishezon, Stable branch curves and braid monodromies, Algebraic geometry (Chicago, Ill., 1980) Lecture Notes in Math., vol. 862, Springer, Berlin-New York, 1981, pp. 107–192. MR 644819
B. Moishezon, The arithmetic of braids and a statement of Chisini, Geometric topology (Haifa, 1992) Contemp. Math., vol. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 151–175. MR 1282761, DOI 10.1090/conm/164/01591
Boris Moishezon and Mina Teicher, Braid group technique in complex geometry. V. The fundamental group of a complement of a branch curve of a Veronese generic projection, Comm. Anal. Geom. 4 (1996), no. 1-2, 1–120. MR 1393558, DOI 10.4310/CAG.1996.v4.n1.a1
Madhav V. Nori, Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 305–344. MR 732347
Mutsuo Oka, Some plane curves whose complements have non-abelian fundamental groups, Math. Ann. 218 (1975), no. 1, 55–65. MR 396556, DOI 10.1007/BF01350067
Francesco Severi, Vorlesungen über algebraische Geometrie (German), Teubner, Leipzig, 1921.
Domingo Toledo, Projective varieties with non-residually finite fundamental group, Inst. Hautes Études Sci. Publ. Math. 77 (1993), 103–119. MR 1249171
Mina Teicher, New invariants for surfaces, Tel Aviv Topology Conference: Rothenberg Festschrift (1998), Contemp. Math., vol. 231, Amer. Math. Soc., Providence, RI, 1999, pp. 271–281. MR 1707349, DOI 10.1090/conm/231/03366
Egbert R. Van Kampen, On the Fundamental Group of an Algebraic Curve, Amer. J. Math. 55 (1933), no. 1-4, 255–260. MR 1506962, DOI 10.2307/2371128
Oscar Zariski, On the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve, Amer. J. Math. 51 (1929), no. 2, 305–328. MR 1506719, DOI 10.2307/2370712
Oscar Zariski, On the irregularity of cyclic multiple planes, Ann. of Math. (2) 32 (1931), no. 3, 485–511. MR 1503012, DOI 10.2307/1968247
Oscar Zariski, On the Topology of Algebroid Singularities, Amer. J. Math. 54 (1932), no. 3, 453–465. MR 1507926, DOI 10.2307/2370887
Oscar Zariski, Algebraic surfaces, Second supplemented edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 61, Springer-Verlag, New York-Heidelberg, 1971. With appendices by S. S. Abhyankar, J. Lipman, and D. Mumford. MR 0469915
References
- J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), no. 2, 275–306. MR 1501429, DOI 10.2307/1989123
- Enrique Artal Bartolo and José Ignacio Cogolludo-Agustín, On the connection between fundamental groups and pencils with multiple fibers, J. Singul. 2 (2010), 1–18. MR 2763015 (2012a:14070), DOI 10.5427/jsing.2010.2a
- Denis Auroux, Symplectic 4-manifolds as branched coverings of $\mathbf {C}\mathbf {P}^2$, Invent. Math. 139 (2000), no. 3, 551–602. MR 1738061 (2000m:53119), DOI 10.1007/s002220050019
- D. Auroux, S. K. Donaldson, and L. Katzarkov, Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves, Math. Ann. 326 (2003), no. 1, 185–203. MR 1981618 (2004c:57039), DOI 10.1007/s00208-003-0418-9
- E. Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math. 12 (1971), 57–61 (German). MR 0293615 (45 \#2692)
- Fabrizio Catanese, Differentiable and deformation type of algebraic surfaces, real and symplectic structures, Symplectic 4-manifolds and algebraic surfaces, Lecture Notes in Math., vol. 1938, Springer, Berlin, 2008, pp. 55–167. MR 2441412, DOI 10.1007/978-3-540-78279-7_2
- Oscar Chisini, Il teorema di esistenza delle trecce algebriche. I, II, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 17 (1954), 143–149, 307–311 (1955) (Italian). MR 0071103 (17,86a)
- J. I. Cogolludo-Agustin and A. Libgober, Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves, J. Reine Angew. Math., to appear.
- P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 0262240 (41 \#6850)
- Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin, 1970 (French). MR 0417174 (54 \#5232)
- 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., vol. 842, Springer, Berlin, 1981, pp. 1–10 (French). MR 636513 (83f:14026)
- Pierre Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273–302 (French). MR 0422673 (54 \#10659)
- Alexandru Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. MR 1194180 (94b:32058), DOI 10.1007/978-1-4612-4404-2
- Alexandru Dimca, Characteristic varieties and logarithmic differential 1-forms, Compos. Math. 146 (2010), no. 1, 129–144. MR 2581244 (2011g:14019), DOI 10.1112/S0010437X09004461
- Alexandru Dimca and Alexander I. Suciu, Which 3-manifold groups are Kähler groups?, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 3, 521–528. MR 2505439 (2011f:32041), DOI 10.4171/JEMS/158
- F. Enriques, Sulla construzione delle funzioni algebriche di due variabili possedenti una data curva diramazione, Ann. Nat. Pure Appl. vol. 1 (1923).
- Robert Friedman and John W. Morgan, Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 27, Springer-Verlag, Berlin, 1994. MR 1288304 (95m:57046)
- William Fulton, On the fundamental group of the complement of a node curve, Ann. of Math. (2) 111 (1980), no. 2, 407–409. MR 569076 (82e:14035), DOI 10.2307/1971204
- Leila Schneps and Pierre Lochak, editors. Geometric Galois actions. 1. Around Grothendieck’s “Esquisse d’un programme”. London Mathematical Society Lecture Note Series, 242. Cambridge University Press, Cambridge, 1997. MR 1483106 (98e:14003), DOI 10.1017/CBO9780511666124
- Joe Harris, On the Severi problem, Invent. Math. 84 (1986), no. 3, 445–461. MR 837522 (87f:14012), DOI 10.1007/BF01388741
- Eriko Hironaka, Alexander stratifications of character varieties, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 2, 555–583. MR 1450425 (98e:14020)
- F. Hirzebruch, Arrangements of lines and algebraic surfaces, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Mass., 1983, pp. 113–140. MR 717609 (84m:14037)
- Vik. S. Kulikov, On Chisini’s conjecture, Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), no. 6, 83–116 (Russian, with Russian summary); English transl., Izv. Math. 63 (1999), no. 6, 1139–1170. MR 1748562 (2001c:14030), DOI 10.1070/im1999v063n06ABEH000267
- Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472 (2005k:14001b), DOI 10.1007/978-3-642-18808-4
- A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J. 49 (1982), no. 4, 833–851. MR 683005 (84g:14030)
- A. Libgober, Alexander invariants of plane algebraic curves, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 135–143. MR 713242 (85h:14017)
- A. Libgober, Characteristic varieties of algebraic curves, computation (Eilat, 2001) NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 215–254. MR 1866902 (2003e:14008)
- Anatoly Libgober, Non vanishing loci of Hodge numbers of local systems, Manuscripta Math. 128 (2009), no. 1, 1–31. MR 2470184 (2010g:32026), DOI 10.1007/s00229-008-0221-8
- Anatoly Libgober and Sergey Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121 (2000), no. 3, 337–361. MR 1761630 (2001j:52032), DOI 10.1023/A:1001826010964
- B. G. Moishezon, Stable branch curves and braid monodromies, Algebraic geometry (Chicago, Ill., 1980) Lecture Notes in Math., vol. 862, Springer, Berlin, 1981, pp. 107–192. MR 644819 (83c:14008)
- B. Moishezon, The arithmetic of braids and a statement of Chisini, Geometric topology (Haifa, 1992) Contemp. Math., vol. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 151–175. MR 1282761 (95d:20069), DOI 10.1090/conm/164/01591
- Boris Moishezon and Mina Teicher, Braid group technique in complex geometry. V. The fundamental group of a complement of a branch curve of a Veronese generic projection, Comm. Anal. Geom. 4 (1996), no. 1-2, 1–120. MR 1393558 (97j:14041)
- Madhav V. Nori, Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 305–344. MR 732347 (86d:14027)
- Mutsuo Oka, Some plane curves whose complements have non-abelian fundamental groups, Math. Ann. 218 (1975), no. 1, 55–65. MR 0396556 (53 \#419)
- Francesco Severi, Vorlesungen über algebraische Geometrie (German), Teubner, Leipzig, 1921.
- Domingo Toledo, Projective varieties with non-residually finite fundamental group, Inst. Hautes Études Sci. Publ. Math. 77 (1993), 103–119. MR 1249171 (94k:14008)
- Mina Teicher, New invariants for surfaces, Tel Aviv Topology Conference: Rothenberg Festschrift (1998), Contemp. Math., vol. 231, Amer. Math. Soc., Providence, RI, 1999, pp. 271–281. MR 1707349 (2001c:14040), DOI 10.1090/conm/231/03366
- Egbert R. Van Kampen, On the Fundamental Group of an Algebraic Curve, Amer. J. Math. 55 (1933), no. 1-4, 255–267. MR 1506962, DOI 10.2307/2371128
- Oscar Zariski, On the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve, Amer. J. Math. 51 (1929), no. 2, 305–328. MR 1506719, DOI 10.2307/2370712
- Oscar Zariski, On the irregularity of cyclic multiple planes, Ann. of Math. (2) 32 (1931), no. 3, 485–511. MR 1503012, DOI 10.2307/1968247
- Oscar Zariski, On the topology of algebroid singularities, Amer. J. Math. 54 (1932), no. 3, 453–465. MR 1507926, DOI 10.2307/2370887
- Oscar Zariski, Algebraic surfaces, Second supplemented edition, with appendices by S. S. Abhyankar, J. Lipman, and D. Mumford, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 61, Springer-Verlag, New York, 1971. MR 0469915 (57 \#9695)
Review Information:
Reviewer:
Anatoly Libgober
Affiliation:
Department of Mathematics,University of Illinois at Chicago, Chicago, Illinois
Email:
libgober@uic.edu
Journal:
Bull. Amer. Math. Soc.
51 (2014), 479-489
DOI:
https://doi.org/10.1090/S0273-0979-2014-01444-2
Published electronically:
March 17, 2014
Additional Notes:
The author was supported during writing this review by a Simons Foundation grant
Review copyright:
© Copyright 2014
American Mathematical Society