Integral points and orbits of endomorphisms on the projective plane

Abstract

We analyze when integral points on the complement of a finite union of curves in P 2 \mathbb {P}^2 are potentially dense. When the logarithmic Kodaira dimension κ ¯ \bar {\kappa } is − ∞ -\infty , we completely characterize the potential density of integral points in terms of the number of irreducible components at infinity and the number of multiple members in a pencil naturally associated to the surface. When κ ¯ = 0 \bar {\kappa } = 0 , we prove that integral points are always potentially dense. The bulk of our analysis concerns the subtle case of κ ¯ = 1 \bar {\kappa }=1 . We determine the potential density of integral points in a number of cases by incorporating the structure theory of affine surfaces and developing an arithmetic framework for studying integral points on surfaces fibered over curves. We also prove, assuming Lang–Vojta’s conjecture, that an orbit under an endomorphism ϕ \phi of P 2 \mathbb {P}^2 can contain a Zariski-dense set of integral points only if there is a nontrivial completely invariant proper Zariski-closed subset of P 2 \mathbb {P}^2 under ϕ \phi . This may be viewed as a generalization of a result of Silverman on P 1 \mathbb {P}^1 .