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

TL;DR

This paper uses the circle method to asymptotically estimate the number of rational linear spaces on hypersurfaces with a fixed discriminant, advancing understanding of rational subvarieties with prescribed geometric properties.

Contribution

It introduces a novel application of the circle method to count linear spaces on hypersurfaces with a specified discriminant, extending previous counting techniques.

Findings

Provides asymptotic estimates for the number of such linear spaces.

Establishes conditions under which the counts are accurate.

Connects the count of rational linear spaces to primitive lattice points.

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.