×

zbMATH — the first resource for mathematics

The fundamental group of a Galois cover of \(\mathbb{C} \mathbb{P}^1 \times T\). (English) Zbl 1037.14006
Let \(T\) be a complex torus in \(\mathbb{C}\mathbb{P}_2\) and \(X= \mathbb{C}\mathbb{P}_1 \times T\). By embedding \(X\) into \( \mathbb{C}\mathbb{P}_5\) by the usual Segre map \( \mathbb{C}\mathbb{P}_1 \times \mathbb{C}\mathbb{P}_2 \rightarrow \mathbb{C}\mathbb{P}_5 \) a generic projection \( f: X \rightarrow \mathbb{C}\mathbb{P}_2 \) is obtained by projecting \(X\) from a general plane in \( \mathbb{C}\mathbb{P}_5 \setminus X \) to \(\mathbb{C}\mathbb{P}_2 \). The Galois cover of \(X\) is the closure of the \(n\)-fibred product \(X_{\text{gal}} = \overline{ X \times_{f} \ldots \times_{f} X - \Delta} \) where \( n = \text{deg}(f) \) and \( \Delta \) is the generalized diagonal.
In this paper the authors compute \( \pi_1( X_{\text{gal}}) = \mathbb{Z}^{10}\) stated as theorem 9.3, using braid monodromy techniques developed by B. Moishezon and M. Teicher [Invent. Math. 89, 601–643 (1987; Zbl 0627.14019)], the van Kampen theorem and various computational methods of groups.

MSC:
14F35 Homotopy theory and fundamental groups in algebraic geometry
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
12F10 Separable extensions, Galois theory
20F36 Braid groups; Artin groups
14E20 Coverings in algebraic geometry
14Q10 Computational aspects of algebraic surfaces
14N05 Projective techniques in algebraic geometry
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
PDF BibTeX XML Cite
Full Text: DOI EMIS EuDML arXiv
References:
[1] M Amram, Galois Covers of Algebraic Surfaces, PhD thesis, Bar-Ilan University (2001)
[2] M Amram, M Teicher, U Vishne, The Coxeter quotient of the fundamental group of a Galois cover of \(\mathbbT\times\mathbbT\), Comm. Algebra 34 (2006) 89 · Zbl 1086.14014
[3] N Bourbaki, Groupes et algèbres de Lie. Chapitres IV-VI, Actualités Scientifiques et Industrielles 1337, Hermann (1968) · Zbl 0186.33001
[4] V S Kulikov, M Taĭkher, Braid monodromy factorizations and diffeomorphism types, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000) 89 · Zbl 1004.14005
[5] B Moishezon, Algebraic surfaces and the arithmetic of braids II, Contemp. Math. 44, Amer. Math. Soc. (1985) 311 · Zbl 0592.14013
[6] B Moishezon, A Robb, M Teicher, On Galois covers of Hirzebruch surfaces, Math. Ann. 305 (1996) 493 · Zbl 0853.14007
[7] B Moishezon, M Teicher, Simply-connected algebraic surfaces of positive index, Invent. Math. 89 (1987) 601 · Zbl 0627.14019
[8] B Moishezon, M Teicher, Braid group technique in complex geometry I: Line arrangements in \(\mathbbC\mathrmP^2\), Contemp. Math. 78, Amer. Math. Soc. (1988) 425 · Zbl 0674.14019
[9] B Moishezon, M Teicher, Finite fundamental groups, free over \(\mathbbZ/c\mathbfZ\), for Galois covers of \(\mathbb{C\mathrmP}^2\), Math. Ann. 293 (1992) 749 · Zbl 0739.14003
[10] B Moishezon, M Teicher, Braid group techniques in complex geometry IV: Braid monodromy of the branch curve \(S_3\) of \(V_3{\rightarrow}\mathbbC\mathrmP^2\) and application to \(\pi_1(\mathbbC\mathrmP^2-S_3,*)\), Contemp. Math. 162, Amer. Math. Soc. (1994) 333 · Zbl 0815.14024
[11] B Moishezon, M Teicher, Braid group technique in complex geometry II: From arrangements of lines and conics to cuspidal curves, Lecture Notes in Math. 1479, Springer (1991) 131 · Zbl 0764.14014
[12] M Teicher, Braid groups, algebraic surfaces and fundamental groups of complements of branch curves, Proc. Sympos. Pure Math. 62, Amer. Math. Soc. (1997) 127 · Zbl 0907.14008
[13] M Teicher, New invariants for surfaces, Contemp. Math. 231, Amer. Math. Soc. (1999) 271 · Zbl 0940.14013
[14] L Rowen, M Teicher, U Vishne, Coxeter covers of the symmetric groups, J. Group Theory 8 (2005) 139 · Zbl 1120.20040
[15] E R V Kampen, On the Fundamental Group of an Algebraic Curve, Amer. J. Math. 55 (1933) 255 · Zbl 0006.41502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.