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

Abstract

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 $F(\mathbf{x})=0$ 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.