Equivariant geometry of odd-dimensional complete intersections of two quadrics

TL;DR

This paper investigates the equivariant birational classification of odd-dimensional complete intersections of two quadrics under finite group actions, linking geometric rationality with Diophantine problems and cohomological invariants.

Contribution

It introduces new methods to analyze equivariant rationality of these varieties, connecting geometric, algebraic, and number-theoretic perspectives.

Findings

Cohomological invariants influence rationality classifications.

Symbol invariants provide new obstructions to equivariant rationality.

Connections established between geometric rationality and Diophantine equations.

Abstract

Fix a finite group $G$. We seek to classify varieties with $G$-action equivariantly birational to a representation of $G$ on affine or projective space. Our focus is odd-dimensional smooth complete intersections of two quadrics, relating the equivariant rationality problem with analogous Diophantine questions over nonclosed fields. We explore how invariants -- both classical cohomological invariants and recent symbol constructions -- control rationality in some cases.