Let T be the complex projective torus, and X the surface ℂℙ1 × T. Let XGal be its Galois cover with respect to a generic projection to ℂℙ2. In this paper we compute the fundamental group of XGal, using the degeneration and regeneration techniques, the Moishezon-Teicher braid monodromy algorithm and group calculations. We show that π1(XGal) = ℤ10.
Amram, Meirav 1 ; Goldberg, David 2 ; Teicher, Mina 3 ; Vishne, Uzi 4
@article{10_2140_agt_2002_2_403,
author = {Amram, Meirav and Goldberg, David and Teicher, Mina and Vishne, Uzi},
title = {The fundamental group of a {Galois} cover of {\ensuremath{\mathbb{C}}\ensuremath{\mathbb{P}}1} {\texttimes} {T}},
journal = {Algebraic and Geometric Topology},
pages = {403--432},
year = {2002},
volume = {2},
number = {1},
doi = {10.2140/agt.2002.2.403},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2002.2.403/}
}
TY - JOUR AU - Amram, Meirav AU - Goldberg, David AU - Teicher, Mina AU - Vishne, Uzi TI - The fundamental group of a Galois cover of ℂℙ1 × T JO - Algebraic and Geometric Topology PY - 2002 SP - 403 EP - 432 VL - 2 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2002.2.403/ DO - 10.2140/agt.2002.2.403 ID - 10_2140_agt_2002_2_403 ER -
%0 Journal Article %A Amram, Meirav %A Goldberg, David %A Teicher, Mina %A Vishne, Uzi %T The fundamental group of a Galois cover of ℂℙ1 × T %J Algebraic and Geometric Topology %D 2002 %P 403-432 %V 2 %N 1 %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2002.2.403/ %R 10.2140/agt.2002.2.403 %F 10_2140_agt_2002_2_403
Amram, Meirav; Goldberg, David; Teicher, Mina; Vishne, Uzi. The fundamental group of a Galois cover of ℂℙ1 × T. Algebraic and Geometric Topology, Tome 2 (2002) no. 1, pp. 403-432. doi: 10.2140/agt.2002.2.403
[1] , Galois Covers of Algebraic Surfaces, PhD thesis, Bar-Ilan University (2001)
[2] , , , The Coxeter quotient of the fundamental group of a Galois cover of $\mathbb{T}\times\mathbb{T}$, Comm. Algebra 34 (2006) 89
[3] , Groupes et algèbres de Lie. Chapitres IV–VI, Actualités Scientifiques et Industrielles 1337, Hermann (1968)
[4] , , Braid monodromy factorizations and diffeomorphism types, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000) 89
[5] , Algebraic surfaces and the arithmetic of braids II, from: "Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982)", Contemp. Math. 44, Amer. Math. Soc. (1985) 311
[6] , , , On Galois covers of Hirzebruch surfaces, Math. Ann. 305 (1996) 493
[7] , , Simply-connected algebraic surfaces of positive index, Invent. Math. 89 (1987) 601
[8] , , Braid group technique in complex geometry I: Line arrangements in $\mathbb{C}\mathrm{P}^2$, from: "Braids (Santa Cruz, CA, 1986)", Contemp. Math. 78, Amer. Math. Soc. (1988) 425
[9] , , Finite fundamental groups, free over $\mathbb{Z}/c\mathbf{Z}$, for Galois covers of $\mathbb{C\mathrm{P}}^{2}$, Math. Ann. 293 (1992) 749
[10] , , Braid group techniques in complex geometry IV: Braid monodromy of the branch curve $S_3$ of $V_3{\rightarrow}\mathbb{C}\mathrm{P}^2$ and application to $\pi_1(\mathbb{C}\mathrm{P}^2-S_3,*)$, from: "Classification of algebraic varieties (L'Aquila, 1992)", Contemp. Math. 162, Amer. Math. Soc. (1994) 333
[11] , , Braid group technique in complex geometry II: From arrangements of lines and conics to cuspidal curves, from: "Algebraic geometry (Chicago, IL, 1989)", Lecture Notes in Math. 1479, Springer (1991) 131
[12] , Braid groups, algebraic surfaces and fundamental groups of complements of branch curves, from: "Algebraic geometry—Santa Cruz 1995", Proc. Sympos. Pure Math. 62, Amer. Math. Soc. (1997) 127
[13] , New invariants for surfaces, from: "Tel Aviv Topology Conference: Rothenberg Festschrift (1998)", Contemp. Math. 231, Amer. Math. Soc. (1999) 271
[14] , , , Coxeter covers of the symmetric groups, J. Group Theory 8 (2005) 139
[15] , On the Fundamental Group of an Algebraic Curve, Amer. J. Math. 55 (1933) 255
Cité par Sources :