ON THE INTEGRAL POINTS ON THE COMPLEMENT OF RAMIFICATION-DIVISORS

Abstract

Following a recent paper by Faltings, we study the integral points on $\bm{P}_2\setminus\mathcal{D}$, where $\mathcal{D}$ is the branch locus of a projection from a surface $\mathcal{X}$; a crucial point in the analysis is that the pull-back of $\mathcal{D}$ in the Galois closure of the projection often splits into several components. As in the paper by Faltings, under certain assumptions we obtain finiteness of the integral points (Theorem 3.1); for instance, we shall find that it suffices if the projection is sufficiently general and if $\mathcal{X}$ has Kodaira number $\ge0$ (Corollary 4.1). We have borrowed freely from Faltings’s paper, for the whole geometrical setting. As to the arithmetic, our method is in part different, relying on the recent paper by Corvaja and Zannier and leading to apparently new conditions. We shall also use a more elementary approach to study a similar situation in arbitrary dimension, where the projection is taken from a hypersurface (Theorem 2.1).

In concrete terms, these results deal with certain diophantine equations $F(x_0,\dots,x_n)=c$.

AMS 2000 Mathematics subject classification: Primary 11D72; 11G35; 11G99