Diophantine Approximation Constants for Varieties over Function Fields

Abstract

By analogy with the program of , we define and study approximation constants for points of a projective variety X defined over K the function field of an irreducible and non-singular in codimension 1 projective variety defined over an algebraically closed field of characteristic zero. In this setting, we use an effective version of Schmidt's subspace theorem, due to J.T.-Y. Wang, to give a sufficient condition for such approximation constants to be computed on a proper K-subvariety of X. We also indicate how our approximation constants are related to volume functions and Seshadri constants.

Hyperbolicity Notions for Varieties Defined over a Non-Archimedean Field

“…It is a fundamental fact that any two embeddings X → P N,an and X → P M,an induce equivalent distances on X(k), see e.g. [Gri15,Proposition 4.3].…”

Divisorial instability and Vojta's Main Conjecture for $\mathbb{Q}$-Fano varieties

“…Similar results hold true in the (characteristic zero) function field setting. Indeed, these viewpoints are well developed in [7] and [9], for example, building on, and applying, earlier results from [19] and [16].…”

On arithmetic inequalities for points of bounded degree

“…1.4. Building on the approach of [13] and [7], for example, we reformulate [14, Conjecture 5.1], expressing it using the language of linear series (see Theorem 3.3). We then adapt the techniques of [13] to show how it can be used to establish a general Arithmetic Second Main Theorem for algebraic points of fixed bounded degree.…”