Braid monodromy factorization for a non-prime K3 surface branch curve

Abstract

Abstract. 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 2 .In [4] we compute the fundamental groups of the complement of the branch curve and of the corresponding Galois cover of the surface. OverviewGiven 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 9 . In order to compute the above groups we degenerate the surface into a union of 16 planes. We then project it onto CP 2 to get a degenerated branch curve S 0 , 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.

On non fundamental group equivalent surfaces

“…1 / 0 and the singular points of .S 1 / 0 , the regeneration process was already done [1], and thus we have the following theorem. …”

“…We denote by X 1 the embedded K3 surface, and by .X 1 / 0 the degenerated surface (see [17] for an explicit definition of a degeneration). The degeneration process has a "local inverse" -the regeneration process (see an explanation in the following subsection), and for it we need to fix a numeration of vertices (and the lines; see Amram-Ciliberto-Miranda-Teicher [1] for details). This is done as shown in Figure 3.…”

“…See [1] for the braid monodromy factorization of the regeneration of our 3-point. In .S 2 / 0 , we have eight points which are 5-point .v 2;j ; 1 Ä j Ä 10; j ¤ 5; 6/ and two points which are 4-point .v 2;j ; j D 5; 6).…”

“…C 2 be the Galois covering corresponding to 1 (see [11] for definitions). Recall that 1 . e X i / D ker ‚ i =h 2 i;j i where ‚ i W 1 .C 2 S i / !…”

Classification of Algebraic Surfaces and Curves

“…This group is the kernel of the projection and it is an important invariant of X. Surfaces which were already investigated are Hirzebruch surfaces, a self product of the projective line and its product with a complex torus, toric varieties and some K3 surfaces [5][6][7][8][9][10][11][12] Having obtained the braid monodromy factorization of the curve S and the fundamental group G, we proceed in two approaches, one based on algebra, the other one based on knot theory.…”