Seminar in Algebraic Geometry and Number Theory

Abstract: Let X be a cubic hypersurface cut out by a non-degenerate rational cubic form in n variables. If n is at least 14, then Heath-Brown has shown that the X has at least one non-trivial rational point. By a result of Kollar, this implies that there are infinitely many rational points on X. In this talk, I will discuss the problem of obtaining quantitative lower bounds for the number of rational points of bounded height on X.