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How often does a cubic hypersurface have a rational point?
Lea Beneish, Christopher Keyes
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A cubic hypersurface in ℙⁿ defined over ℚ is given by the vanishing locus of a cubic form f in n+1 variables. It is conjectured that when n ≥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 ℙⁿ, ordered by the height of f, with a rational point for n ≥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 ℙⁿ for n ≥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 ℙ³ with a rational point.
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