Counting rational points of a Grassmannian

We prove an estimate on the number of rational points on the Grassmannian variety of bounded twisted height, refining the classical results of Schmidt (Duke Math J 35:327–339, 1968) and Thunder (Compos Math 88(2):155–186, 1993) over the rational field: most importantly, our formula counts all points. Among the consequences are a couple of new implications on the classical subject of counting rational points on flag varieties.