Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties

2013/06/30 by David McKinnon, Mike Roth · 57 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Algebraic number #Algebraic variety #Ample line bundle #Commutative Algebra and Its Applications #Diophantine approximation #Diophantine equation #Discrete mathematics #Invariant (physics) #Line bundle #Mathematical analysis #Mathematical physics #Mathematics #Polynomial and algebraic computation #Projective variety #Pure mathematics #math.AG #math.NT #msc:11G50 #msc:11J87 #msc:14G05 #msc:14G40

paper · pdf · doi:10.1007/s00222-014-0540-1

published in Inventiones mathematicae 200(2), 513-583 (Springer Science+Business Media) · 55 pages, published version

openalex publication_date 2014/08/21 · arxiv created 2015/04/27 · arxiv updated 2015/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper, we associate an invariant α_x(L) to an algebraic point x on an algebraic variety X with an ample line bundle L. The invariant α measures how well x can be approximated by rational points on X, with respect to the height function associated to L. We show that this invariant is closely related to the Seshadri constant ε_x(L) measuring local positivity of L at x, and in particular that Roth's theorem on P¹ generalizes as an inequality between these two invariants valid for arbitrary projective varieties.

Citations

• Diophantine Geometry
• Lectures on Arakelov Geometry
• Algebraic Number Theory
• Positivity in Algebraic Geometry I
• Asymptotic cohomological functions on projective varieties