Rational points on algebraic varieties

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This project will focus on the study of rational points on algebraic varieties. Given an algebraic variety over a number field, natural questions are: Is there a rational point? If yes, are there infinitely many? If also yes, can one obtain a finer quantitative description of the distribution of the rational points? These problems are very difficult in general, but in this project the aim is to make some progress for some special classes of varieties (in particular solve some new cases of Manin's conjecture). It also expects to contribute to a popular current research theme, that of considering these problems in families, such as studying the distribution of varieties in a family with a rational point, or controlling failures of the Hasse principle in families. To solve such problems one normally uses a combination of techniques from algebraic geometry, algebraic number theory and analytic number theory.