×

On the number of rational points of bounded height on smooth bilinear hypersurfaces in biprojective space. (English) Zbl 1020.11046

The Fano variety \(X\subset {\mathbb P}_{\mathbb Q}^n\times {\mathbb P}_{\mathbb Q}^n\) given by the vanishing of a suitably general bihomogeneous form of bidegree \((1,1)\) is isomorphic over \({\mathbb Q}\) to the flag variety of lines in hyperplanes. It is embedded in \({\mathbb P}^N\) by \(|-K_X|=|{\mathcal O}(n,n)|\): a choice of basis for this linear system determines a height function \(H({\mathbf p})=\prod_\wp \max\limits_i |p_i|_\wp\), where \({\mathbf p}=(p_0:\ldots:p_N)\) and \(\wp\) runs over all places of \({\mathbb Q}\). The purpose of the present paper is to re-prove by a new method the result, due to J. Franke, Yu. Manin and Yu. Tschinkel [Invent. Math. 95, 421–435 (1989; Zbl 0674.14012)] and to J. L. Thunder [Compos. Math. 88, 155–186 (1993; Zbl 0806.11030)] that the number of rational points of \(X\) of height at most \(B\) grows like \(B^n\log B\) (for \(n\geq 3\)).
The geometric part of the method consists of the observations that \(X\) may in practice be replaced by the flag variety \(Y\) in the Segre embedding, and that (following Schanuel, Salberger and Peyre) one may count integral points in the affine cone on \(Y\) rather than rational points on \(Y\) itself. The bulk of the paper is taken up with analytic number theory, namely an estimate of the number of integral points obtained by means of Heath-Brown’s variant of the circle method. Similar methods should in principle be applicable to other hypersurfaces in products of projective spaces.

MSC:

11G50 Heights
14G05 Rational points
11P55 Applications of the Hardy-Littlewood method
11D99 Diophantine equations
PDFBibTeX XMLCite
Full Text: DOI