Base Loci of Linear Series and Diophantine Approximation

University Of New Mexico, Albuquerque NM

Investigators

Abstract

From the most abstract concepts of theoretical physics to the cryptosystems used by millions of people every day (often without realizing it) , algebraic geometry and number theory have a genuine and vital presence in twenty-first century life. This proposal seeks to develop tools of algebraic geometry with a view toward number theoretic applications. Of central importance is the problem of finding rational or whole number solutions to polynomial equations, the most famous example of which is Fermat's Last Theorem recently established by Wiles. The new geometric techniques developed under this proposal will be applied to limit the number of rational solutions of a more general polynomial equation. Many situations in diophantine approximation produce a linear series on a smooth projective variety. Often there are arguments employing vanishing theorems and intersection theory that force the series to move, showing the base locus to be empty. On the other hand, varous hypothetical assumptions-for example, assuming that there are infinitely many rational points on a curve of genus at least two, or assuming that there are good rational approximations of an algebraic irrational number-- force the linear series in question to have a non-empty base locus. The resulting contradiction shows the hypothetical assumption to be false. This line or reasoning has been used to establish Mordell's conjecture and Roth's theorem respectively, and will be applied in this project to study higher dimensional problems, such as the Schmidt subspace theorem and Faltings' theorem on rational points of subvarieties of abelian varieties, with a long term goal of extending and or strengthening these theorems. The key algebro-geometric background required in these arguments is a detailed study of numerical properties of the base locus of a linear series. In particular, a key question is whether or not Seshadri constants control the existence of global sections of a line bundle even when the bundle is not positive.