A counterexample to the geometric Chevalley-Warning conjecture

TL;DR

This paper constructs a specific quartic threefold with particular singularities that exhibits torsion in its middle homology, providing a counterexample to a longstanding conjecture in algebraic geometry.

Contribution

It presents the first known counterexample to the geometric Chevalley-Warning conjecture involving a quartic threefold with L-rational singularities and torsion homology.

Findings

Constructed a quartic threefold with L-rational singularities

Demonstrated torsion in the middle homology group of the threefold

Answered a question of Brown and Schnetz for all fields of characteristic zero

Abstract

We construct a quartic threefold with L-rational singularities which has torsion in its middle homology group. This answers a question of Brown and Schnetz for all fields of characteristic zero.