Vojta’s conjecture on rational surfaces and the abc conjecture

Abstract

AbstractWe prove Vojta’s conjecture for some rational surfaces. Moreover, for similar but different rational surfaces, we show that their Vojta’s conjecture is related to the abc conjecture. More specifically, we prove that Vojta’s conjecture on these surfaces implies a special case of the abc conjecture, while the abc conjecture implies Vojta’s conjecture on these surfaces. The argument carries over to the holomorphic case, so we unconditionally obtain Griffiths’ conjecture for the same situation. To prove these results, we prove and use some properties of Farey sequences.