Manin's conjecture for a certain singular cubic hypersurface
DOI:
https://doi.org/10.2140/mscand.2026.132.161Keywords:
Manin's conjecture, singular cubic hypersurfacesAbstract
Let $\mathcal{S}_Q$ denote the hypersurface in $\mathbb{P}_{\mathbb{Q}}^{m+1}$ defined by $x^3= Q(y_1, \ldots, y_m)z$, where $Q$ is a primitive positive definite quadratic form in $m$ variables with integral coefficients. That variety ranges over a class of singular cubic hypersurfaces as $Q$ varies. For $\mathcal{S}_Q$, we prove that
(i) Manin's conjecture is true if $Q$ is locally determined with $2\mid m$ and $m\geq 4$;
(ii) in general, Manin's conjecture holds for $\mathcal{S}_Q$ up to a leading constant if $2\mid m$ and $m\geq 6$ under some technical assumptions.
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