How often does a cubic hypersurface have a rational point?
How often does a cubic hypersurface have a rational point?
A cubic hypersurface in $\mathbb{P}^n$ defined over $\mathbb{Q}$ is given by the vanishing locus of a cubic form $f$ in $n+1$ variables. It is conjectured that when $n \geq 4$, such cubic hypersurfaces satisfy the Hasse principle. This is now known to hold on average due to recent work of …