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arXiv · 1703.06148

Manin's conjecture for a class of singular cubic hypersurfaces

Jianya Liu·Jie Wu·Yongqiang Zhao
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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.

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Jianya Liu, Jie Wu, Yongqiang Zhao. 2017-03-17. Manin's conjecture for a class of singular cubic hypersurfaces. https://arxiv.org/abs/1703.06148

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