Vojta's Conjecture on Multiple Blowups of ℙ² and the abc conjecture

2016/01/15 by Yu Yasufuku, Yasufuku, Yu
Mathematics · #11B57 #11J97 #14G25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1601.03825

openalex publication_date 2016/01/15 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

Related

• Vojta's abc Conjecture for algebraic tori and applications over function fields
• The exceptional set in Vojta's conjecture for algebraic points of bounded degree
• Vojta's abc conjecture for entire curves in toric varieties highly ramified over the boundary
• A complex case of Vojta's general abc conjecture and cases of Campana's orbifold conjecture
• Vojta's conjecture for singular varieties