Height of rational points on random Fano hypersurfaces (Q2027511)

The paper under review investigates how often a rational point of small height can lie on a Fano hypersurface of large height. The author proves that for fixed dimension and degree, that almost all Fano hypersurfaces contain no rational points of height much smaller than the height of the hypersurface itself. Specifically, the author computes the following limit for fixed degree $d$ and dimension $n$ of $V$, provided that $n\geq d$, $d\geq 2$, and $(n,d)\neq(2,2)$): $$\lim_{A\to\infty}\frac{\#\{V\text{ of height at most }A\text{ and a rational point of height at most }\psi(A)\}}{\#\{V \text{ of height at most } A\}}=1,$$ where $\psi$ is any positive real function of order $o(A)$. The author proves analogous results in the case $(d,n)=(2,2)$, and a version of the celebrated Batyrev-Manin conjecture proven on average over all Fano hypersurfaces of fixed degree and dimension. Specifically, the author proves that the average number of rational points of height at most $B$ on a Fano hypersurface of degree $d$ and dimension $n$ (for $n\geq d\geq 2$ and $(n,d)\neq(2,2)$) is on the order of $B/A$, provided that $A\geq B^{1/(n+1-d)}$. (The result in the paper is more precise -- we omit the exact statement in the interest of brevity.)