Asymptotics of Integral Points on a log Fano Variety

Abstract

In this thesis we study integral points of bounded height on three log Fano threefolds, following the paper \textit{Integral Points of Bounded Height on a log Fano Threefold} by Florian Wilsch. We parametrize the integral points on the log Fano threefolds using the universal torsor method and obtain lattice points satisfying certain (coprimality) conditions. With the height function induced by log-anticanonical bundles on the threefolds, we bound the integral points, leading to three counting functions. To obtain asymptotic formulae for two of the counting functions, we apply Möbius inversion and we replace sums by integrals. We show that this method cannot be extended in a straightforward way to the third counting function and instead we determine an upper bound.