How often does a cubic hypersurface have a rational point?

Abstract

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 Browning, Le Boudec, and Sawin. Using this result, we determine the proportion of cubic hypersurfaces in $\mathbb{P}^n$, ordered by the height of $f$, with a rational point for $n \geq 4$ explicitly as a product over primes $p$ of rational functions in $p$. In particular, this proportion is equal to 1 for cubic hypersurfaces in $\mathbb{P}^n$ for $n \geq 9$; for $100\%$ of cubic hypersurfaces, this recovers a celebrated result of Heath-Brown that non-singular cubic forms in at least 10 variables have rational zeros. In the $n=3$ case, we give a precise conjecture for the proportion of cubic surfaces in $\mathbb{P}^3$ with a rational point.