The Hasse principle for lines on diagonal surfaces

Abstract

Given a number field $k$ and a positive integer $d$, in this paper we consider the following question: does there exist a smooth diagonal surface of degree $d$ in ${\mathbb{P}}^3$ over $k$ which contains a line over every completion of $k$, yet no line over $k$? We answer the problem using Galois cohomology, and count the number of counter-examples using a result of Erdős.

14 pages

References in corpus (3)

• On the destruction of minimal foliations
• Forms representing forms and linear spaces on hypersurfaces
• Counting lines on surfaces