Cheng GONG, Wan-Yuan XU, Mina TEICHER, Meirav AMRAM

Fundamental groups of Galois covers of degree 5 surfaces

Let X be an algebraic surface of degree 5, which is considered a branch cover of CP2 with respect to a generic projection. The surface has a natural Galois cover with Galois group S5 . In this paper, we deal with the fundamental groups of Galois covers of degree 5 surfaces that degenerate to nice plane arrangements; each of them is a union of five planes such that no three planes meet in a line

PDF

___

[1] Amram M, Ciliberto C, Miranda R, Teicher M. Braid monodromy factorization for a non-prime K3 surface branch curve. Israel Journal of Mathematics 2009; 170: 61-93.
[2] Amram M, Dettweiler M, Friedman M, Teicher M. Fundamental group of the complements of the Cayley’s singularities. Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry 2009; 50 (2): 469- 482.
[3] Amram M, Friedman M, Teicher M. The fundamental group of the complement of the branch curve of the second Hirzebruch surface. Topology 2009; 48: 23-40.
[4] Amram M, Friedman M, Teicher M. The fundamental group of the branch curve of the complement of the surface CP1 × T . Acta Mathematica Sinica 2009; 25 (9): 1443-1458.
[5] Amram M, Garber D, Shwartz R, Teicher M. 8-point – regenerations and applications. Advances in Geometric Analysis, Advanced Lectures in Mathematics (ALM)-series, International Press of Boston, in cooperation with the Higher Education Press (Beijing) and the Stefan Banach International Mathematical Centre, ALM 2012; 21: 307-342.
[6] Amram M, Goldberg D, Teicher M, Vishne U. The fundamental group of a Galois cover of CP1 × T . Algebraic & Geometric Topology 2002; 2: 403-432.
[7] Amram M, Goldberg D. Higher degree Galois covers of CP1 × T . Algebraic & Geometric Topology 2004; 4: 841-859.
[8] Amram M, Lehman R, Shwartz R, Teicher M. Classification of fundamental groups of Galois covers of surfaces of small degree degenerating to nice plane arrangements. Topology of algebraic varieties and singularities, Contemporary Mathematics 2011; 538: 63-92, American Mathematical Society, Providence, RI 2011.
[9] Amram M, Ogata S. Toric varieties–degenerations and fundamental groups. The Michigan Mathematical Journal 2006; 54: 587-610.
[10] Amram M, Teicher M. The fundamental group of the complement of the branch curve of T × T in C 2 . Osaka Journal of Math. 2003; 40: 1-37.
[11] Amram M, Teicher M. On the degeneration, regeneration and braid monodromy of T × T . Acta Applicandae Mathematicae 2003; 75: 195-270.
[12] Amram M, Teicher M, Vishne U. The fundamental group of the Galois cover of the surface T × T . International Journal of Algebra and Computation 2008; 18 (8): 1259-1282.
[13] Artin E. Theory of braids. Annals of Mathematics 1947; 48: 101-126.
[14] Auroux D, Donaldson S, Katzarkov L, Yotov M. Fundamental groups of complements of plane curves and symplectic invariants. Topology 2004; 43: 1285-1318.
[15] Calabri A, Ciliberto C, Flamini F, Miranda R. On the K2 of degenerations of surfaces and the multiple point formula. Annals of Mathematics 2007; 165: 335-395.
[16] Catanese F. On the moduli spaces of surfaces of general type. Journal of Differential Geometry 1984; 19: 483-515.
[17] Catanese F. (Some) old and new results on algebraic surfaces. In: Joseph A, Mignot F, Murat F, Prum B, Rentschler R (editors) First European Congress of Mathematics. Progress in Mathematics, vol 3. Birkhäuser Basel.
18] Miyaoka Y. On the Chern numbers of surfaces of general type. Inventiones Mathematicae 1977; 42: 225–237.
[19] Cayley A. A memoir on cubic surfaces. Philosophical Transactions of the Royal Society London 1869; 159: 231-326.
[20] Chisini O. Sulla identitá birationale delle funzioni algebriche di due variabili dotate di una medesima curva di diramazione. Rend. Istit. Lombardo 1944; 77: 339-356 (in Italian).
[21] Ciliberto C, Lopez A, Miranda R. Projective degenerations of K3 surfaces, Gaussian maps, and Fano threefolds. Inventiones Mathematicae 1993; 114: 641-667.
[22] Friedman M, Teicher M. On fundamental groups related to the Hirzebruch surface F1 . Science China Information Sciences A. 2008; 51: 728-745.
[23] Friedman M, Teicher M. The regeneration of a 5-point. Pure and Applied Math. Quarterly 2008; 4 (2): 383-425.
[24] Friedman M, Teicher M. On fundamental groups related degeneratable surfaces: conjectures and examples. Annali della Scuola normale superiore di Pisa. 2012; XI (5): 565-603.
[25] Gieseker D. Global moduli for surfaces of general type. Inventiones Mathematicae 1977; 41: 233-282.
[26] Gonçalves DL, Guaschi J. The braid groups of the projective plane. Algebraic & Geometric Topology 2004; 4: 757-780.
[27] Liedtke C. Fundamental groups of Galois closures of generic projections. Transactions of the AMS 2010; 362: 2167-2188.
[28] Kulikov V. On Chisini’s conjecture. Izvestiya: Mathematics. 1999; 63 (6): 1139-1170.
[29] Kulikov V. On Chisini’s conjecture, II. Izvestiya: Mathematics. 2008; 72 (5): 901-913.
[30] Moishezon B, Robb A, Teicher M. On Galois covers of Hirzebruch surfaces. Mathematische Annalen 1996; 305: 493-539.
[31] Moishezon B, Teicher M. Galois covers in theory of algebraic surfaces. Proceedings of Symposia in Pure Math. 1987; 46: 47-65.
[32] Moishezon B, Teicher M. Simply connected algebraic surfaces of positive index. Inventiones Mathematicae 1987; 89: 601-643.
[33] Moishezon B, Teicher M. Braid group technique in complex geometry I, Line arrangements in CP2 . Contemporary Mathematics 1988; 78: 425-555.
[34] Moishezon B, Teicher M. Braid group technique in complex geometry II, From arrangements of lines and conics to cuspidal curves. Algebraic Geometry, Lecture Notes in Mathematics 1991; 1479: 131-180.
[35] Moishezon B, Teicher M. Braid group technique in complex geometry III: Projective degeneration of V3 . Contemporary Mathematics 1994; 162: 313-332.
[36] Moishezon B, Teicher M. Braid group technique in complex geometry IV: Braid monodromy of the branch curve S3 of V3 → CP2 and application to π1(C 2 − S3, ∗). Contemporary Mathematics 1994; 162: 332-358.
[37] Moishezon B,Teicher M. Braid group technique in complex geometry V: The fundamental group of a complement of a branch curve of a Veronese generic projection. Communications in Analysis and Geometry 1996; 4: 1-120.
[38] Salmon G. A Treatise on Analytic Geometry of Three Dimensions. 5th ed. Volume 1. London, UK: Longmans, Green and Company, 1912.
[39] Schläfli L. On the distribution of surfaces of third order into species, in reference to the absence or presence of singular points, and the reality of their lines. Philosophical Transactions of the Royal Society London 1863; 153: 193-241.
[40] Teicher M. New invariants for surfaces. Tel Aviv Topology Conference: Rothenberg Festschrift 1998; 271-281, Contemporary Mathematics 1999; 231, American Mathematical Society, Providence, RI, 1999.
[41] van Kampen ER. On the fundamental group of an algebraic curve. American Journal of Mathematics 1933; 55: 255-260.
[42] Zariski O. On the topological discriminant group of a Riemann surface of genus p. American Journal of Mathematics 1937; 59: 335-358.