On the volume of a six-dimensional polytope

Abstract

This note is a comment to the paper by D.R.Heath-Brown and B.Z.Moroz (Math Proc. Camb. Phil. Soc. 125 (1999)). That paper concerns with the projective surface $S$ in ${\mathbb{P}}^3$ defined by the equation $x_1{x}_2{x}_3=x_4^3$. It is shown there that the evaluation of the leading term of the asymptotic formula for the number of rational points of bounded height in $S(\Q)$ is equivalent to the evaluation of the volume of some 6-dimensional polytope $\P$. The volume of $\P$ is known from several papers; we calculate this volume by elementary method using symmetry of the polytope $\P$. We also discuss a combinatorial structure of $\P$.