Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties

Expectations, Concave Transforms, Chow weights, and Roth's theorem for varieties

We explain how complexity of rational points on projective varieties can be interpreted via the theories of Chow forms and Okounkov bodies. Precisely, we study discrete measures on filtered linear series and build on work of Boucksom and Chen, Boucksom-et-al and Ferretti. For example, we show how asymptotic volume and Seshadri constants are related to the theories of Chow weights and Okounkov bodies for very ample linear series. We also consider arithmetic applications within the context of Diophantine approximation. In this direction, we establish Roth-type theorems which can be expressed in …

A generalized Schmidt subspace theorem for closed subschemes

We prove a generalized version of Schmidt's subspace theorem for closed subschemes in general position in terms of suitably defined Seshadri constants with respect to a fixed ample divisor. Our proof builds on previous work by Evertse and Ferretti, Corvaja and Zannier, and others, and uses standard techniques from algebraic geometry such as notions of positivity, blowing-ups and direct image sheaves. As an application, we recover a higher-dimensional Diophantine approximation theorem of K.F. Roth-type due to D. McKinnon and M. Roth with a significantly shortened proof, while simultaneously ext…

A geometric formulation of Siegel's diophantine theorem

In this paper, we introduce an algebro-geometric formulation for Siegel's theorem using an improvement of Lang's version of Roth's theorem over finitely generated fields of characteristic zero. In fact, we prove that, for an affine open curve in an irreducible smooth curve of genus at least one, any finitely generated subgroup of the additive group of the affine ambient space intersects the open curve in only finitely many points. This was proved only for finitely generated subgroups defined over a localization of the ring of integers of a number field by Mahler and others.

Diophantine approximation constants for varieties over function fields

By analogy with the program of McKinnon-Roth, 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 …

二面体轨道产生的无理Seshadri常数

Irrational Seshadri constants from dihedral orbits

本文在九个非常一般点的爆破$X_9$上构造了无理Seshadri常数$2\sqrt{5}$，通过证明$\mathbb{P}^1\times\mathbb{P}^1$上$(1,1)$类在十点处的常数为$1/\sqrt{5}$，并利用二面体轨道及形变方法完成证明。

Self-Similar Fractals and Arithmetic Dynamics

The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as `similarity' maps. Self-similar fractals are subsets of algebraic varieties which can be written as a finite and disjoint union of `similar' copies. Fractals provide a framework in which, one can unite some results and conjectures in Diophantine geometry. We define a well-behaved notion of dimension for self-similar fractals. We also prove a fractal version of Roth's theorem for algebraic points on a variety approximated by elements of a fractal subset. As a cons…