Geometric invariants and Geometric consistency of Manin’s conjecture

Abstract

Manin’s conjeture states that the asymptotic growth of the number of rational points

on a Fano variety over a number field is governed by certain geometric invariants (a

and b-constants). In this thesis we study the behaviour of these geometric invariants

and show that Manin’s conjecture is geometrically consistent. In the first part, we

study the behaviour of the b-constant in families and show that the b-constant is

constant on very general fibers of a family of algebraic varieties. If the fibers of the

family are uniruled, then we show that the b-constant is constant on general fibers.

In the second part, we study the behaviour of the a-constant (Fujita invariant) under

pull-back to generically finite covers and prove a conjecture of Lehmann-Tanimoto

about finiteness of covers. In the last part, based on joint work with B. Lehmann

and S. Tanimoto, we prove geometric consistency of Manin’s conjecture by showing

that the rational points contributed by subvarieties or covers with larger geometric

invariants are contained in a thin set.