On fundamental groups related to degeneratable surfaces: conjectures and examples
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, pp. 565-603.

We argue that for a smooth surface S, considered as a ramified cover over ℂℙ 2 , branched over a nodal-cuspidal curve Bℂℙ 2 , one could use the structure of the fundamental group of the complement of the branch curve π 1 (ℂℙ 2 -B) to understand other properties of the surface and its degeneration and vice-versa. In this paper, we look at embedded-degeneratable surfaces — a class of surfaces admitting a planar degeneration with a few combinatorial conditions imposed on its degeneration. We close a conjecture of Teicher on the virtual solvability of π 1 (ℂℙ 2 -B) for these surfaces and present two new conjectures on the structure of this group, regarding non-embedded-degeneratable surfaces. We prove two theorems supporting our conjectures, and show that for ℂℙ 1 ×C g , where C g is a curve of genus g, π 1 (ℂℙ 2 -B) is a quotient of an Artin group associated to the degeneration.

Published online:
Classification: 14D06, 14Q10, 14H20, 14H30, 20F36
Friedman, Michael 1; Teicher, Mina 1

1 Department of Mathematics Bar-Ilan University 52900 Ramat Gan, Israel
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Friedman, Michael; Teicher, Mina. On fundamental groups related to degeneratable surfaces: conjectures and examples. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, pp. 565-603. http://archive.numdam.org/item/ASNSP_2012_5_11_3_565_0/

[1] M. Amram, D. Goldberg, M. Teicher and U. Vishne, The fundamental group of a Galois cover of ℂℙ 1 ×𝕋, Algebr. Geom. Topol. 2 (2002), 403–432. | EuDML | MR | Zbl

[2] M. Amram and D. Goldberg,Higher degree Galois covers of ℂℙ 1 ×𝕋, Algebr. Geom. Topol. 4 (2004), 841–859. | EuDML | MR | Zbl

[3] M. Amram, M. Teicher and U. Vishne, The Coxeter quotient of the fundamental group of a Galois cover of 𝕋×𝕋, Comm. Algebra 34 (2006), 89–106. | MR | Zbl

[4] M. Amram, R. Shwartz and M. Teicher, Coxeter covers of the classical Coxeter groups, Int. J. Algebra Comput 20 (2010), 1041–1062. | MR | Zbl

[5] M. Amram and O. Shoetsu, Degenerations and fundamental groups related to some special Toric varieties, Michigan Math. J. 54 (2006), 587–610. | MR | Zbl

[6] M. Amram, M. Friedman and M. Teicher, The fundamental group of the complement of the branch curve of the second Hirzebruch surface, Topology 48 (2009), 23–40. | MR | Zbl

[7] M. Amram, M. Friedman and M. Teicher, The fundamental group of the complement of the branch curve of ℂℙ 1 ×𝕋, Acta Mathematica Sin. (Engl. Ser.) 25 (2009), 1443–1458. | MR | Zbl

[8] D. Auroux, S. K. Donaldson, L. Katzarkov and M. Yotov, Fundamental groups of complements of plane curves and symplectic invariants, Topology 43 (2004), 1285–1318. | MR | Zbl

[9] A. Calabri, C. Ciliberto, F. Flamini and R. Miranda, Degenerations of scrolls to unions of planes, Atti Accad. Naz. Licei, Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 17 (2006), 95–123. | MR | Zbl

[10] A. Calabri, C. Ciliberto, F. Flamini and R. Miranda, On the K 2 of degenerations of surfaces and the multiple point formula, Ann. of Math. 165 (2007), 335–395. | MR | Zbl

[11] A. Calabri, C. Ciliberto, F. Flamini and R. Miranda, On degenerations of surfaces, arXiv:math/0310009v2 [math.AG] (2008). | Numdam | MR

[12] M. Friedman and M. Teicher, The regeneration of a 5-point, Pure Appl. Math. Q. 4 (2008) (Fedor Bogomolov special issue, part I), 383–425. | MR | Zbl

[13] M. Friedman and M. Teicher, On non fundamental group equivalent surfaces, Algebr. Geom. Topol. 8 (2008) 397–433. | MR | Zbl

[14] M. Friedman and M. Teicher, On fundamental groups related to the Hirzebruch surface F 1 , Sci. China Ser. A. 51 (2008), 1–18. | MR | Zbl

[15] R. Hartshorne, “Families of Curves in ℂℙ 3 and Zeuthen’s Problem”, Mem. Amer. Math. Soc. 130, 1997, no. 617. | MR | Zbl

[16] J. E. Humphreys, “Reflection Groups and Coxeter Groups”, Cambridge University Press, 1992. | MR | Zbl

[17] Vik. Kulikov, On Chisini’s conjecture, Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), 83–116. | MR | Zbl

[18] Vik. Kulikov, On Chisini’s conjecture II, Izv. Math. 72 (2008). | MR | Zbl

[19] V. S. Kulikov and Vik.S. Kulikov, On complete degenerations of surfaces with ordinary singularities in 3 , Mat. Sb. 201 (2010), 129–160. | MR | Zbl

[20] C. Liedtke, On fundamental groups of Galois closures of generic projections, Trans. Amer. Math. Soc. 362 (1010), 2167–2188. arXiv:0505.406. | MR | Zbl

[21] B. Moishezon, Algebraic surfaces and the arithmetic of braids. II, In: “Combinatorial Methods in Topology and Algebraic Geometry”, Rochester, NY, (1982), Contemp. Math., Vol. 44, Amer. Math. Soc., Providence, RI, (1985) 311–344. | MR | Zbl

[22] B. Moishezon, On cuspidal branch curves, J. Algebraic Geom. 2 (1993), 309–384. | MR | Zbl

[23] B. Moishezon, A. Robb and M. Teicher, On Galois covers of Hirzebruch surfaces, Math. Ann. 305 (1996), 493–539. | EuDML | MR | Zbl

[24] B. Moishezon and M. Teicher, Galois covers in theory of algebraic surfaces, In: “Algebraic Geometry Bowdoin, 1985”, Proc. Sympos. Pure Math., Vol. 46, Part 2, Amer. Math. Soc., Providence, RI, 1987, pp. 47–65. | MR | Zbl

[25] B. Moishezon and M. Teicher, Simply connected algebraic surfaces of positive index, Invent. Math. 89 (1987), 601–643. | EuDML | MR | Zbl

[26] B. Moishezon and M. Teicher, Braid group technique in complex geometry, I: Line arrangements in ℂℙ 2 , Contemp. Math. 78 (1988), 425–555. | MR | Zbl

[27] B. Moishezon and M. Teicher, Braid group technique in complex geometry, II: From arrangements of lines and conics to cuspidal curves, In: “Algebraic Geometry”, Lecture Notes in Math., Vol. 1479, 1990, 131–180. | MR | Zbl

[28] B. Moishezon and M. Teicher, Braid group techniques in complex geometry III: Projective degeneration of V 3 , Contemp. Math. 162 (1994), 313–332. | MR | Zbl

[29] B. Moishezon and M. Teicher, Braid group techniques in complex geometry IV: Braid monodromy of the branch curve S 3 of V 3 ℂℙ 2 and application to π 1 :(ℂℙ 2 -S 3 ,*), Contemp. Math. 162 (1993), 332–358. | MR | Zbl

[30] B. Moishezon and M. 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), 1–120. | MR | Zbl

[31] A. Robb, “The Topology of Branch Curves of Complete Intersections”, Doctoral Thesis, Columbia University, 1994. | MR

[32] A. Robb, On branch curves of algebraic surfaces, In: “Singularities and Complex Geometry” (Bijug, 1994), AMS/IP Stud. Adv. Math. 5, Amer. Math. Soc., Providence, RI, 1997, pp. 193–221. | MR | Zbl

[33] L. Rowen, M. Teicher and U. Vishne, Coxeter covers of the symmetric groups, J. Group Theory 8 (2005), 139–169. | MR | Zbl

[34] M. Teicher, New invariants for surfaces, Contemp. Mathematics 231 (1999), 271–281. | MR | Zbl

[35] M. Teicher, On the quotient of the braid group by commutators of transversal half-twists and its group actions, Topology Appl. 78 (1997), 153–186. | MR | Zbl

[36] G. Zappa, Sulla degenerazione delle superficie algebriche in sistemi di piani distinti, con applicazioni allo studio delle rigate, Atti R. Accad. d’Italia, Mem. Cl. Sci. FF., MM. e NN. 13 (1943), 989–1021. | MR | Zbl

[37] O. Zariski, On the topological discriminant group of a Riemann surface of genus p, Amer. J. Math. 59 (1937), 335–358. | JFM | MR