# Manin's conjecture for a class of singular cubic hypersurfaces

## TL;DR

This paper proves Manin's conjecture for a specific class of singular cubic hypersurfaces and provides an asymptotic count of rational points of bounded height on these varieties, advancing understanding of rational point distribution on singular cubic hypersurfaces.

## Contribution

It establishes the first asymptotic formula for rational points on certain singular cubic hypersurfaces not covered by classical theorems, confirming Manin's conjecture for these cases.

## Key findings

- Proves Manin's conjecture for the hypersurfaces $S_n$.
- Provides an asymptotic formula for the number of rational points.
- Advances the study of rational points on singular cubic hypersurfaces.

## Abstract

Let $n$ be a positive multiple of $4$. We establish an asymptotic formula for the number of rational points of bounded height on singular cubic hypersurfaces $S_n$ defined by $$ x^3=(y_1^2 + \cdots + y_n^2)z . $$ This result is new in two aspects: first, it can be viewed as a modest start on the study of density of rational points on those singular cubic hypersurfaces which are not covered by the classical theorems of Davenport or Heath-Brown; second, it proves Manin's conjecture for singular cubic hypersurfaces $S_n$ defined above.