Braid Monodromy Factorization for a Non-prime K3 Surface Branch Curve Amram Meirav, Ciliberto Ciro, Miranda Rick and Teicher Mina

In this paper we consider a non-prime K3 surface of degree 16, and study a specific degeneration of it, known as the (2, 2)-pillow degeneration, [10]. We study also the braid monodromy factorization of the branch curve of the surface with respect to a generic projection onto CP. In [4] we compute the fundamental groups of the complement of the branch curve and of the corresponding Galois cover of the surface. 1. Overview Given a projective surface and a generic projection to the plane, the fundamental group of the complement of the branch curve is one of its most important invariants. Our goal is to compute this group and the fundamental group of the Galois cover (which is known to be a certain quotient of the fundamental group of the complement of the branch curve, see [13]). This goal is achieved in [4]. In this paper we deal with a non-prime K3 surface of degree 16 which is embedded in P. In order to compute the above groups we degenerate the surface into a union of 16 planes. We then project it onto CP to get a degenerated branch curve S0, from which one can compute the braid monodromy factorization and the branch curve S of the projection of the original K3 surface. With this information we will be able to apply the van Kampen Theorem (see [20]) and the regeneration rules (see [18]) to get presentations for the relevant fundamental groups. The idea of using degenerations for these purposes appears already in [3], [9], [11], [13] and [17]. Degenerations of K3 surfaces, of which the one we use here is an example, were constructed in [10] and are called pillow degenerations. Partially supported by the DAAD fellowship (Germany), the Golda Meir postdoctoral fellowship (the Einstein mathematics institute, Hebrew university, Jerusalem), the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center ”Group Theoretic Methods in the Study of Algebraic Varieties” of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099). Date: May 18, 2008.