Equivariant geometry of odd-dimensional complete intersections of two quadrics

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.

Equivariant Burnside groups and toric varieties

“…There are many similarities but also subtle distinctions between these points of view, highlighted, e.g., in [20]. One of the similarities is that the cohomological obstruction (1.1) applies also as an obstruction to (stable) linearizability of the G-action, since for linear actions of G, the invariant (1.1) vanishes (see, e.g., [3,Prop.…”

Torsors and stable equivariant birational geometry

“…• equivariant enhancements of intermediate Jacobians and cycle invariants [HT21]; • equivariant Burnside groups [KT20], [KT21a].…”

Cohomology of finite subgroups of the plane Cremona group

“…We approach the computation of H 1 (G, Pic(X)) via the Brauer group Br([X/G]) of the quotient stack [X/G], or more classically, the equivariant Brauer group, introduced in [13], see also [15,Sect. 2.3].…”