Curves of low genus on surfaces and applications to Diophantine problems

Abstract

We describe in detail a technique due to Vojta for finding the explicit set of curves of low genus on certain algebraic surfaces of general type, and refine some of its aspects. We then provide applications of this method to three Diophantine problems.

We prove under the Bombieri-Lang Conjecture that there are finitely many non-trivial sequences of integers of length 11 whose squares have constant second differences, and we prove unconditionally the analogous result for function fields of characteristic zero.

We prove under the Bombieri-Lang Conjecture that there are finitely many integer sequences of length 8 whose k-th powers have second differences equal to 2, we give an unconditional result for function fields of characteristic zero. Moreover, this gives new examples of surfaces having no curves of genus 0 or 1.

The third application is related to the surface parametrizing perfect cuboids. We give some new properties about their curves of genus 0 or 1 and we give new bounds for the degree of curves in this surface, in terms of their genus.