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.

In the first part of this thesis, we study the local fundamental group of low-dimensional log canonical singularities. In dimensions 2 and 3 we establish some constraints on the possible local fundamental groups, while in dimensions 3 and 4 we construct examples of interesting groups that can appear. In dimension 2, by classifying all the possible singularities, we can prove that the local fundamental groups are virtually solvable. Moreover, we give a bound on the number of generators and relations of the group, along with identifying the circumstances under which this maximum is achieved. In dimension 3, we show that free groups with 2 or more generators do not appear as local fundamental groups of log canonical isolated singularities. In dimensions 3 and 4, we construct the fundamental groups of 2-dimensional closed manifolds and some special 3-dimensional manifolds, respectively, as the local fundamental groups of isolated log canonical singularities. One notable example includes the connected sum of copies of S1 × S2, leading to free groups appearing in dimension 4. In the second part, we study the orbifold fundamental groups of the smooth locus of Calabi-Yau type pairs of low coregularity. Here, we establish that the fundamental group of pairs with low coregularity exhibits similar behavior to log Calabi-Yau pairs of low dimension, specifically being virtually abelian of bounded rank. Furthermore, we prove in the case of virtually nilpotency, there are effective bounds on the index and length depending only on the dimension and coregularity.