On non fundamental group equivalent surfaces

Mina Teicher, Michael Friedman

Research output: Contribution to journalArticlepeer-review

Abstract

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).

Original languageEnglish
Pages (from-to)397-433
Number of pages37
JournalAlgebraic and Geometric Topology
Volume8
Issue number1
DOIs
StatePublished - 2008
Externally publishedYes

Keywords

  • Branch curve
  • Curves and singularities
  • Fundamental group
  • Generic projection

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