Approximation Constants for Closed Subschemes of Projective Varieties

Abstract

Diophantine approximation is a branch of number theory with a long history, going back at least to the work of Dirichlet and Liouville in the 1840s. The innocent-looking question of how well an arbitrary real algebraic number can be approximated by rational numbers (relative to the size of the denominator of the approximating rational number) took more than 100 years to resolve, culminating in the definitive Fields Medal-winning work of Klaus Roth in 1955.

Much more recently, David McKinnon and Mike Roth have re-phrased and generalized this Diophantine approximation question to apply in the setting of approximating algebraic points on projective varieties defined over number fields. To do this, they defined an "approximation constant", depending on the point one wishes to approximate and a given line bundle. This constant measures the tradeoff between the closeness of the approximation and the arithmetic complexity of the point used to make the approximation, as measured by a height function associated to the line bundle.

In particular, McKinnon and Roth succeeded in proving lower bounds on the approximation constant in terms of the "Seshadri constant" associated to the given point and line bundle, measuring local positivity of the line bundle around the point. Appropriately interpreted, these results generalize the classical work of Liouville and Roth, and the corresponding McKinnon-Roth theorems are therefore labelled "Liouville-type" and "Roth-type" results.

Recent work of Grieve and of Ru-Wang have taken the Roth-type theorems even further; in contrast, we explore results of Liouville-type, which are more elementary in nature. In Chapter 2, we lay the groundwork necessary to define the approximation constant at a point, before generalizing the McKinnon-Roth definition to approximations of arbitrary closed subschemes. We also introduce the notion of an essential approximation constant, which ignores unusually good approximations along proper Zariski-closed subsets. After verifying that our new approximation constant truly does generalize the constant of McKinnon-Roth, Chapter 3 establishes a fundamental lower bound on the approximation constants of closed subschemes of projective space, depending only on the equations cutting out the subscheme.

In Chapter 4, we provide a series of explicit computations of approximation constants, both for subschemes satisfying suitable geometric conditions, and for curves of low degree in projective 3-space. We will encounter difficulties computing the approximation constant exactly for general cubic curves, and we spend some time showing why some of the more evident approaches do not succeed. To conclude the chapter, we take up the question of large gaps between the ordinary and essential approximation constants, by considering approximations to a certain rational point on a diagonal quartic surface. Finally, in Chapter 5, we generalize the Liouville-type results of McKinnon-Roth.