Hasse principle for intersections of two quadrics via Kummer surfaces

TL;DR

This paper establishes new cases of the Hasse principle for certain algebraic surfaces, including Kummer surfaces and intersections of two quadrics, under specific finiteness assumptions related to Tate--Shafarevich groups.

Contribution

It proves the Hasse principle for Kummer surfaces from 2-coverings of Jacobians of genus 2 curves and for certain quartic del Pezzo surfaces, extending known results.

Findings

Hasse principle holds for Kummer surfaces under Tate--Shafarevich finiteness

Hasse principle applies to smooth intersections of two quadrics in high dimensions

Results depend on finiteness assumptions of Tate--Shafarevich groups

Abstract

We prove new cases of the Hasse principle for Kummer surfaces constructed from 2-coverings of Jacobians of genus 2 curves, assuming finiteness of relevant Tate--Shafarevich groups. Under the same assumption, we deduce the Hasse principle for quartic del Pezzo surfaces with trivial Brauer group and irreducible or completely split characteristic polynomial, hence the Hasse principle for smooth complete intersections of two quadrics in the projective space of dimension at least 5.