Rational points on twisted K3 surfaces and derived equivalences

TL;DR

The paper constructs examples of twisted K3 surfaces over various fields that are derived equivalent but differ in the existence of rational points, answering a recent open question.

Contribution

It provides explicit examples of derived equivalent twisted K3 surfaces with differing rational point properties, using Hassett--Várilly-Alvarado's construction.

Findings

Existence of derived equivalent twisted K3 surfaces over $\\mathbb{Q}$, $\\mathbb{Q}_2$, and $\mathbb{R}$

Examples where one surface has a rational point and the other does not

Negation of a recent question by Hassett and Tschinkel

Abstract

Using a construction of Hassett--V\'arilly-Alvarado, we produce derived equivalent twisted K3 surfaces over $\mathbb{Q}$, $\mathbb{Q}_2$, and $\mathbb{R}$, where one has a rational point and the other does not. This answers negatively a question recently raised by Hassett and Tschinkel.