We determine an asymptotic formula for the number of integral points of bounded height on a blow-up of ${\mathbb {P}}^3$ outside certain planes using universal torsors.

We determine an asymptotic formula for the number of integral points of bounded height on a certain toric variety, which is incompatible with part of a preprint by Chambert-Loir and Tschinkel. We provide an alternative interpretation of the asymptotic formula we get. To do so, we construct an analogue of Peyre's constant α and describe its relation to a new obstruction to the Zariski density of integral points in certain regions of varieties.

We determine an asymptotic formula for the number of integral points of bounded height on a blow-up of P 3 outside certain planes using universal torsors.

In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type A 1 + A 3 and prove an analogue of Manin's conjecture for integral points with respect to its singularities and its lines.

In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type $\mathbf {A}_1+\mathbf {A}_3$ and prove an analogue of Manin’s conjecture for integral points with respect to its singularities and its lines.