A family of varieties with exactly one pointless rational fiber

TL;DR

This paper constructs a specific family of algebraic varieties over a rational base, demonstrating a case with exactly one fiber lacking rational points, thus illustrating a particular phenomenon in rational point distribution.

Contribution

It provides an explicit example of a family with exactly one fiber without rational points, building on Poonen's earlier construction.

Findings

Explicit example of a family with one pointless rational fiber

Demonstrates the distribution of rational points in algebraic families

Connects to Poonen's theoretical construction

Abstract

We construct a concrete example of a 1-parameter family of smooth projective geometrically integral varieties over an open subscheme of P^1_Q such that there is exactly one rational fiber with no rational points. This makes explicit a construction of Poonen.