On the number of linear spaces on hypersurfaces with a prescribed discriminant

For a given form $F\in \mathbb{Z}[x_1,\dots ,x_s]$ we apply the circle method in order to give an asymptotic estimate of the number of $m$-tuples $\mathbf x_1, \dots, \mathbf x_m$spanningalinearspaceonthehypersurface$F(\mathbf x) = 0$withthepropertythat$\det ( (\mathbf x_1, \dots, \mathbf x_m)^t , (\mathbf x_1, \dots, \mathbf x_m)) = b$. This allows us in some measure to count rational linear spaces on hypersurfaces whose underlying integer lattice is primitive.