ar X iv : a lg - g eo m / 9 70 30 17 v 2 2 6 Fe b 19 99 Chern Classes of Fibered Products of Surfaces

In this paper we introduce a formula to compute Chern classes of fibered products of algebraic surfaces. For f : X → CP a generic projection of an algebraic surface, we define Xk for k ≤ n (n = deg f) to be the closure of k products of X over f minus the big diagonal. For k = n (or n − 1), Xk is called the full Galois cover of f w.r.t. full symmetric group. We give a formula for c 1 and c2 of Xk. For k = n the formulas were already known. We apply the formula in two examples where we manage to obtain a surface with a high slope of c 1 /c2. We pose conjectures concerning the spin structure of fibered products of Veronese surfaces and their fundamental groups. 0. Introduction. When regarding an algebraic surface X as a topological 4-manifold, it has the Chern classes c1, c2 as topological invariants. These Chern classes satisfy: c1, c2 ≥ 0 5c1 ≥ c2 − 36 Signature = τ = 1 3 (c1 − 2c2) The famous Bogomolov-Miyaoka-Yau inequality from 1978 (see [Re], [Mi], [Y]) states that the Chern classes of an algebraic surface also satisfy the inequality