Rational lines on cubic hypersurfaces II

TL;DR

This work sharpens the bound for the existence of $K$-rational lines on cubic hypersurfaces over number fields by developing an adaptive non-real quadratic-extension strategy. A central technical advance shows that for a diagonal quadratic form in $n=2k+1$ variables there exists a non-real extension $L/K$ making the form vanish on an affine $L$-linear space of dimension at least $k$, enabling a reduction to a smaller-dimensional search for a line. Applying this to cubics yields: if $n\ge 35$ in general, $n\ge 33$ for imaginary quadratic $K$, or $n\ge 31$ when $K=\mathbb{Q}$, then $C=0$ contains a $K$-rational line, improving Wooley's bound; the paper also proves sharpness of the dimension bound and discusses a Leep-based variant for imaginary quadratic fields. The results extend Birch–Wooley type rational-space conclusions to broader settings and demonstrate a robust local-to-global framework for quadratic forms under quadratic extensions.

Abstract

We show that any rational cubic hypersurface of dimension at least 33 defined over a number field $K$ vanishes on a $K$-rational projective line, reducing the previous lower bound of Wooley by two. For $K=\mathbb Q$ we can reduce the bound to 29. The main ingredients are a result on linear spaces on quadratic forms over suitable non-real quadratic field extensions, and recent work of Bernert and Hochfilzer on cubic forms over imaginary quadratic number fields for the rational case.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 2.1
• proof
• Lemma 2.2
• proof