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

Yu Yasufuku

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

Number Theory 2016-01-26 v2 Algebraic Geometry

Authors: Yu Yasufuku

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.

Keywords

Vojta's conjecture abc conjecture rational surfaces blowups of P^2 Farey sequences Diophantine geometry

Cite

@article{arxiv.1601.03825,
  title  = {Vojta's Conjecture on Multiple Blowups of $\mathbb{P}^2$ and the $abc$ conjecture},
  author = {Yu Yasufuku},
  journal= {arXiv preprint arXiv:1601.03825},
  year   = {2016}
}

Comments

22 pages; abstract modified to make it absolutely clear that this paper does not prove the abc conjecture

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

Abstract

Keywords

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