On the density of rational lines on diagonal cubic hypersurfaces

Abstract

In this paper, we establish the asymptotic estimates for the rational lines on diagonal cubic hypersurfaces defined by ${\sum }_{i=1}^s{c}_i{x}_i^3=0$ with $c_i\in \mathbb{Z}\setminus \{0\},$ provided that $s\ge 19.$ This improves the previously known bound $s\ge 21$ required to obtain such asymptotic estimates. Our approach develops a multidimensional shifting variables argument together with a pruning argument, and exploits the recent progress on the Parsell-Vinogradov system.

14 pages, all comments welcome!