Chern classes of fibered products of surfaces

Abstract

In this paper we introduce a formula to compute Chern classes of fibered products of algebraic surfaces. For $f:X\to {\mathbb{CP}}^2$ a generic projection of an algebraic surface, we define $X_k$ for $k\le n(n=\mathrm{deg}f)$ to be the closure of $k$ products of $X$ over $f$ minus the big diagonal. For $k=n$ (or $n-1),X_k$ is called the full Galois cover of $f$ w.r.t. full symmetric group. We give a formula for $c_1^2$ and $c_2$ of $X_k.$ 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 $c_1^2/c_2.$ We pose conjectures concerning the spin structure of fibered products of Veronese surfaces and their fundamental groups.