A conjecture of Manin’s relates the number of rational points of bounded height on Fano varieties to geometric properties. Analogously to this conjecture on rational points, we study the distribution of integral points of bounded height on certain varieties, by adapting the torsor method to this setting.
An asymptotic formula determined this way for a certain toric variety partially contradicts an unpublished preprint by Chambert-Loir and Tschinkel. This discrepancy can be explained by an obstruction to the existence of integral points on a part of this variety. Moreover, this obstruction is related to some constants – analogous to Peyre’s alpha for rational points – that appear in asymptotic formulas for the number of integral points of bounded height.