The Fundamental Group of the Complement of the Branch Curve of T × T in C

This paper is the second in a series of three papers concerning the surface T×T , where T is a complex torus. We compute the fundamental group of the branch curve of the surface in C, using the van Kampen Theorem and the braid monodromy factorization of the curve. 1. Background The concentration on the fundamental group of a complement of a branch curve of an algebraic surfaceX with respect to a generic projection onto CP, leads us to the computation of π1(XGal) the fundamental group of the Galois cover of X with respect to this generic projection. Galois covers are surfaces of a general type. Bogomolov conjectured that the Galois covers corresponding to generic projections of algebraic surfaces to CP have infinite fundamental groups. In [7] we justify Bogomolov’s conjecture by proving that π1(T ×T )Gal is an infinite group. In order to compute π1(T × T )Gal, we have to enclode the braid monodromy factorization of the branch curve S of T × T . Then we have to apply the van Kampen Theorem on the factors in the factorization in order to get relations for π1(C 2 − S, ∗) the fundamental group of the complement of S in C. The fundamental group of the Galois cover XGal is known to be a quotient of a certain subgroup of the fundamental group of the complement of S. We recall shortly the computations from [6]. Let X = T × T be an algebraic surface (where T is a complex torus) embedded in CP, and f : X → CP be a generic projection. We degenerate X to a union of 18 planes X0([6, Section 3]). We numerate the lines and vertices as shown in Figure 1. Date: February 1, 2008. 1 2 MEIRAV AMRAM AND MINA TEICHER