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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 \({\mathbf {x}}_1, \dots , {\mathbf {x}}_m\) spanning a linear space on the hypersurface \(F(X) = 0\) with the property that \(\det ( ({\mathbf {x}}_1, \dots , {\mathbf {x}}_m)^t \, ({\mathbf {x}}_1, \dots , {\mathbf {x}}_m)) = b\). This allows us in some measure to count rational linear spaces on hypersurfaces whose underlying integer lattice is primitive.

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