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On non fundamental group equivalent surfaces
Mina Teicher and Michael Friedman
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Algebraic & Geometric Topology 8
(2008) 397–433
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Abstract
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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).
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Keywords
fundamental group, generic projection,
curves and singularities, branch curve
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Mathematical Subject Classification
Primary: 14H30, 14J28
Secondary: 14F35, 14H20, 14Q05, 20F36,
57M12
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Publication
Received: 1 August 2007
Revised: 27 December 2007
Accepted: 28 December 2007
Published: 12 May 2008
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