Calculating the fundamental group of Galois cover of the (2,3)-embedding of \(\mathbb{CP}^1 \times T\). (English) Zbl 1477.14086
In the paper under review, the authors investigate the fundamental group of the Galois cover of the \((2,3)\)-embedding of \(X:=\mathbb{P}^{1}_{\mathbb{C}} \times T\), where \(T\) is a complex torus in \(\mathbb{P}^{2}_{\mathbb{C}}\). More precisely, since \(\mathbb{P}^{1}_{\mathbb{C}}\) can be embedded into \(\mathbb{P}^{m}_{\mathbb{C}}\) with \(m\geq 2\) and \(T\) can be embedded into \(\mathbb{P}^{n-1}_{\mathbb{C}}\) with \(n\geq 3\), then \(\mathbb{P}^{1}_{\mathbb{C}} \times T\) can be embedded into a larger projective space by the Segre embedding of \(\mathbb{P}^{m}_{\mathbb{C}} \times \mathbb{P}^{n-1}_{\mathbb{C}}\). Such an embedding is the \((m,n)\)-embedding of \(\mathbb{P}^{1}_{\mathbb{C}} \times T\) and one denotes the image of the embedding by \(X_{m,n}\). The resulting surface is of general type with index zero. In order to investigate the fundamental group, the authors apply Moishezon-Teicher’s algorithm combined with certain degeneration techniques.
Reviewer: Piotr Pokora (Kraków)
MSC:
14N20 | Configurations and arrangements of linear subspaces |
14F35 | Homotopy theory and fundamental groups in algebraic geometry |
14J29 | Surfaces of general type |
14Q10 | Computational aspects of algebraic surfaces |
20F36 | Braid groups; Artin groups |
32S22 | Relations with arrangements of hyperplanes |
52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |
14H20 | Singularities of curves, local rings |
References:
[1] | Amram, M.; Ciliberto, C.; Miranda, R., Braid monodromy factorization for a non-prime K3 surface branch curve, Israel Journal of Mathematics, 170, 61-93 (2009) · Zbl 1205.14050 · doi:10.1007/s11856-009-0020-2 |
[2] | Amram, M.; Friedman, M.; Teicher, M., The fundamental group of the branch curve of the complement of the surface ℂℙ^1 × T, Acta Mathematica Sinica, 25, 9, 1443-1458 (2009) · Zbl 1178.14018 · doi:10.1007/s10114-009-6473-8 |
[3] | Amram, M.; Goldberg, D.; Teicher, M., The fundamental group of a Galois cover of ℂℙ^1 × T, Algebraic & Geometric Topology, 2, 403-432 (2002) · Zbl 1037.14006 · doi:10.2140/agt.2002.2.403 |
[4] | Amram, M.; Goldberg, D., Higher degree Galois covers of ℂℙ^1 × T, Algebraic & Geometric Topology, 4, 841-859 (2004) · Zbl 1069.14065 · doi:10.2140/agt.2004.4.841 |
[5] | Amram, M.; Shwartz, R.; Teicher, M., Coxeter covers of the classical Coxeter groups, International Journal of Algebra and Computation, 20, 1041-1062 (2010) · Zbl 1237.20031 · doi:10.1142/S0218196710006023 |
[6] | Amram, M.; Teicher, M., The fundamental group of the complement of the branch curve of T × T in ℂ^2, Osaka Journal of Math., 40, 1-37 (2003) |
[7] | Amram, M.; Teicher, M., On the degeneration, regeneration and braid monodromy of T × T, Acta Appl. Math., 75, 195-270 (2003) · Zbl 1085.14504 · doi:10.1023/A:1022396230382 |
[8] | Artin, E., Theory of braids, Ann. Math., 48, 101-126 (1947) · Zbl 0030.17703 · doi:10.2307/1969218 |
[9] | Friedman, M.; Teicher, M., The regeneration of a 5-point, Pure and Applied Math. Quarterly, 4, 2, 383-425 (2008) · Zbl 1168.14022 · doi:10.4310/PAMQ.2008.v4.n2.a5 |
[10] | Goncalves, D. L.; Guaschi, J., The braid groups of the projective plane, Algebraic & Geometric Topology, 4, 757-780 (2004) · Zbl 1056.20024 · doi:10.2140/agt.2004.4.757 |
[11] | Kulikov, V.; Teicher, M., Braid monodromy factorizations and diffeomorphism types, Izv. Math., 64, 2, 311-341 (2000) · Zbl 1004.14005 · doi:10.1070/IM2000v064n02ABEH000285 |
[12] | Moishezon, B.; Teicher, M., Galois covers in theory of algebraic surfaces, Proceedings of Symposia in Pure Math., 46, 47-65 (1987) · Zbl 0644.14012 · doi:10.1090/pspum/046.2/927973 |
[13] | Moishezon, B.; Teicher, M., Simply connected algebraic surfaces of positive index, Invent. Math., 89, 601-643 (1987) · Zbl 0627.14019 · doi:10.1007/BF01388987 |
[14] | Moishezon, B.; Teicher, M., Braid group technique in complex geometry I, Line arrangements in ℂℙ2, Contemporary Math., 78, 425-555 (1988) · Zbl 0674.14019 · doi:10.1090/conm/078/975093 |
[15] | Moishezon, B.; Teicher, M., Braid group technique in complex geometry II, From arrangements of lines and conics to cuspidal curves, Algebraic Geometry, Lect. Notes in Math., 1479, 131-180 (1991) · Zbl 0764.14014 · doi:10.1007/BFb0086269 |
[16] | Moishezon, B.; Teicher, M., Braid group technique in complex geometry III: Projective degeneration of V3, Contemp. Math., 162, 313-332 (1994) · Zbl 0815.14023 · doi:10.1090/conm/162/01540 |
[17] | Moishezon, B.; Teicher, M., Braid group technique in complex geometry IV: Braid monodromy of the branch curve S3 of V3 → ℂℙ^2 and application to π_1(ℂ2 - S_3,*), Contemporary Math., 162, 332-358 (1994) |
[18] | Rowen, L.; Teicher, M.; Vishne, U., Coxeter covers of the symmetric groups, J. Group Theory, 8, 139-169 (2005) · Zbl 1120.20040 · doi:10.1515/jgth.2005.8.2.139 |
[19] | Van Kampen, E. R., On the fundamental group of an algebraic curve, Amer. J. Math., 55, 255-260 (1933) · Zbl 0006.41502 · doi:10.2307/2371128 |
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