Rational points of bounded height on Fano varieties

Points of Bounded Height on Algebraic Varieties

Introduction 1 1. Heights on the projective space 3 1.1. Basic height function 3 1.2. Height function on the projective space 5 1.3. Behavior under maps 7 2. Heights on varieties 9 2.1. Divisors 9 2.2. Heights 13 3. Conjectures 19 3.1. Zeta functions and counting 19 3.2. Height zeta function 20 3.3. Results and methods 22 3.4. Examples 24 4. Compactifications of Semi-Simple Groups 26 4.1. A Con...

Rational points on varieties

Rational Points on Some Fano Quadratic Bundles

We study the number of rational points of bounded height on a certain threefold. The accumulating subvarieties are Zariski-dense in this example. The computations support an extension of a conjecture of Manin to this situation.

Rational Connectedness of Log Q-fano Varieties

Let X be a log Q-Fano variety, i.e, there exists an effective Q-divisor D such that (X,D) is Kawamata log terminal (klt) and −(KX + D) is nef and big. By a result of Miyaoka-Mori [15], X is uniruled. The conjecture ([10], [16]) predicts that X is rationally connected. In this paper, apply the theory of weak (semi) positivity of (log) relative dualizing sheaves f∗(KX/Y + ∆) (which has been devel...

Counting points of fixed degree and bounded height on linear varieties

We count points of fixed degree and bounded height on a linear projective variety defined over a number field k. If the dimension of the variety is large enough compared to the degree we derive asymptotic estimates as the height tends to infinity. This generalizes results of Thunder, Christensen and Gubler and special cases of results of Schmidt and Gao.