The Hasse principle for lines on diagonal surfaces

TL;DR

This paper investigates whether certain diagonal surfaces over number fields contain lines locally everywhere but not globally, using Galois cohomology and combinatorial counting, revealing new counterexamples to the Hasse principle.

Contribution

It introduces a Galois cohomology approach to analyze the existence of lines on diagonal surfaces and counts counterexamples using Erdős's result.

Findings

Existence of diagonal surfaces with local lines but no global line

Quantitative count of counterexamples using combinatorial methods

Application of Galois cohomology to rational points on 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 Erd\H{o}s.