On non fundamental group equivalent surfaces

Teicher, Mina¹ ; Friedman, Michael¹

¹Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel.

Algebraic and Geometric Topology, Tome 8 (2008) no. 1, pp. 397-433

Résumé

In this paper we present an example of two polarized K3 surfaces which are not Fundamental Group Equivalent (their fundamental groups of the complement of the branch curves are not isomorphic; denoted by FGE) but the fundamental groups of their related Galois covers are isomorphic. For each surface, we consider a generic projection to ℂℙ2 and a degenerations of the surface into a union of planes – the “pillow" degeneration for the non-prime surface and the “magician" degeneration for the prime surface. We compute the Braid Monodromy Factorization (BMF) of the branch curve of each projected surface, using the related degenerations. By these factorizations, we compute the above fundamental groups. It is known that the two surfaces are not in the same component of the Hilbert scheme of linearly embedded K3 surfaces. Here we prove that furthermore they are not FGE equivalent, and thus they are not of the same Braid Monodromy Type (BMT) (which implies that they are not a projective deformation of each other).

Détail

DOI : 10.2140/agt.2008.8.397

Keywords: fundamental group, generic projection, curves and singularities, branch curve

Bibliographie

[1] M Amram, C Ciliberto, R Miranda, M Teicher, Braid monodromy factorization for a non-prime $K3$ surface branch curve

[2] M Amram, M Friedman, M Teicher, The fundamental group of complement of a branch curve of a Hirzebruch surface $F_{2,(2,2)}$

[3] M Amram, M Teicher, The fundamental group of the complement of the branch curve of the double torus, Journal of Mathematics 40 (2003) 587

[4] D Auroux, S K Donaldson, L Katzarkov, M Yotov, Fundamental groups of complements of plane curves and symplectic invariants, Topology 43 (2004) 1285

[5] C Ciliberto, R Miranda, On the Gaussian map for canonical curves of low genus, Duke Math. J. 61 (1990) 417

[6] C Ciliberto, R Miranda, M Teicher, Pillow degenerations of $K3$ surfaces, from: "Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001)", NATO Sci. Ser. II Math. Phys. Chem. 36, Kluwer Acad. Publ. (2001) 53

[7] M Friedman, M Teicher, On the fundamental group related to the Hirzebruch surface $F_1$

[8] V S Kulikov, M Teicher, Braid monodromy factorizations and diffeomorphism types, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000) 89

[9] E Liberman, M Teicher, The Hurwitz equivalence problem is undecidable

[10] C Liedtke, On fundamental groups of Galois closures of generic projections, Bonner Mathematische Schriften [Bonn Mathematical Publications], 367, Universität Bonn Mathematisches Institut (2004)

[11] B Moishezon, On cuspidal branch curves, J. Algebraic Geom. 2 (1993) 309

[12] B Moishezon, A Robb, M Teicher, On Galois covers of Hirzebruch surfaces, Math. Ann. 305 (1996) 493

[13] B Moishezon, M Teicher, Simply-connected algebraic surfaces of positive index, Invent. Math. 89 (1987) 601

[14] B Moishezon, M Teicher, Braid group technique in complex geometry. I. Line arrangements in $\mathbb{CP}^2$, from: "Braids (Santa Cruz, CA, 1986)", Contemp. Math. 78, Amer. Math. Soc. (1988) 425

[15] B Moishezon, M Teicher, 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

[16] B Moishezon, M Teicher, Braid group techniques in complex geometry. IV. Braid monodromy of the branch curve $S_3$ of $V_3\to \mathbb{CP}^2$ and application to $\pi: (\mathbb{CP}^2-S_3,*)$, from: "Classification of algebraic varieties (L'Aquila, 1992)", Contemp. Math., Amer. Math. Soc. (1994) 333

[17] B Moishezon, 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

[18] A Robb, The topology of branch curves of complete intersections, PhD thesis, Columbia university (1994)

[19] A Robb, On branch curves of algebraic surfaces, from: "Singularities and complex geometry (Beijing, 1994)", AMS/IP Stud. Adv. Math. 5, Amer. Math. Soc. (1997) 193

[20] E R Van Kampen, On the Fundamental Group of an Algebraic Curve, Amer. J. Math. 55 (1933) 255