Vojta's Conjecture on Multiple Blowups of $\mathbb{P}^2$ and the $abc$ conjecture

TL;DR

This paper explores the deep connection between Vojta's conjecture and the $abc$ conjecture on certain rational surfaces, establishing equivalences and unconditional cases using properties of Farey sequences.

Contribution

It demonstrates the equivalence between Vojta's conjecture and the $abc$ conjecture for specific rational surfaces and proves Vojta's conjecture unconditionally for related surfaces.

Findings

Vojta's conjecture implies a special case of the $abc$ conjecture.

The $abc$ conjecture implies Vojta's conjecture on these surfaces.

Unconditional proof of Vojta's conjecture for certain rational surfaces.

Abstract

We show that Vojta's conjecture for some rational surfaces 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. Moreover, for similar but different rational surfaces, we prove Vojta's conjecture unconditionally. To prove these results, we use some (possibly new) properties of Farey sequences.