This project focuses on problems at the interface of algebraic geometry and number theory, concerning connections between global geometric invariants of algebraic varieties and rational points. Among specific problems to be addressed are: effective computations of Brauer-Manin obstructions on K3 surfaces, the study of potential density of rational and integral points on higher-dimensional varieties over number fields and function fields, and the study of their asymptotic distribution. One of the key issues is the investigation of rational curves on these varieties and their deformations.
Arithmetic geometry studies integral solutions of polynomial equations in several variables with integral coefficients from a geometric point of few. One of the simplest questions is: find rectangular triangles with all three sides integers. This leads to the study of rational points on a circle of radius 1. Higher-dimensional geometric objects display a very high degree of complexity; finding and describing rational points on them poses tremendous theoretical and computational challenges. Advances in arithmetic geometry have important applications in the area of information transmission and storage, cryptography and graph theory.
This project focused on several major problems in higher-dimensional arithmetic geometry. This is a relatively new field of mathematics at the interface of geometry, i.e. the study of shapes, and arithmetic, the study of integral solutions of algebraic equations with integral coefficients. Geometry and arithmetic are the main pillars of mathematics, with origins in ancient times, but only recently it was recognized that there are tight links between these fields: algebraic equations determine geometric objects, and their global geometric invariants, such as curvature, seem to strongly influence the set of integral solutions of these equations. The simplest geometric objects are curves, e.g., a projective line and its images in higher-dimensional varieties, called rational curves. In the last three decades, there has been tremendous progress in the understanding of arithmetic properties of curves. On the other hand, the arithmetic of higher-dimensional varieties remains still largely unexplored. Recent advances in the Minimal Model Program and the study of curves and their moduli spaces have given new strong impulses for the development of higher-dimensional arithmetic geometry. The project fits into this context: its key objective was to uncover new connections between arithmetic properties of higher-dimensional algebraic varieties, e.g., the existence of rational and integral points, and the geometry of spaces of rational curves on the underlying variety. Specifically, the goal was to understand the reasons for: - existence of rational and integral points, - uniformity in the distribution of points, - observed connections between rational points and the geometry of spaces of rational curves. Among the principal results are: 1. Proof of effective computability of obstructions to the existence of rational points, in joint work with B. Hassett and A. Kresch, 2. Proof of a version of the "Birational section conjecture", joint with F. Bogomolov, 3. Description of spaces of sections of low-degree surface fibrations over the projective line, joint with B. Hassett, 4. Description of "extremal" rational curves on holomorphic symplectic varieties, with B. Hassett. These results have already stimulated subsequent work by graduate students and postdoctoral fellows at the Courant Institute. The theoretical advances in arithmetic geometry outlined above could soon lead to new developments in areas ranging from cryptography to mathematical physics.
