Manin's conjecture for singular cubic hypersurfaces

TL;DR

This paper investigates Manin's conjecture for a class of singular cubic hypersurfaces defined by equations involving quadratic forms, proving the conjecture in specific cases and up to a leading constant in others.

Contribution

It establishes the validity of Manin's conjecture for singular cubic hypersurfaces defined by quadratic forms under certain conditions on the dimension and divisibility.

Findings

Manin's conjecture holds when m is even and m ≥ 4 for locally determined Q.

The conjecture is valid up to a leading constant when m is even and m ≥ 6.

Results extend understanding of rational points on singular cubic hypersurfaces.

Abstract

Let S Q denote x 3 = Q(y 1 ,. .. , y m)z where Q is a primitive positive definite quadratic form in m variables with integer coefficients. This S Q ranges over a class of singular cubic hypersurfaces as Q varies. For S Q we prove (i) Manin's conjecture is true if Q is locally determined with 2 | m and m 4; (ii) in general Manin's conjecture is true up to a leading constant if 2 | m and m 6.