Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 4

Special issue celebrating the work of Herb Clemens

Guest Editor: Ron Donagi

Equivariant geometry of odd-dimensional complete intersections of two quadrics

Pages: 1555 – 1597

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n4.a8

Authors

Brendan Hassett (Department of Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Yuri Tschinkel (Courant Institute, New York University, New York, N.Y., U.S.A.; and Simons Foundation, New York, N.Y., U.S.A.)

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.

Keywords

equivariant geometry, rationality constructions, complete intersections of two quadrics

2010 Mathematics Subject Classification

Primary 14E08. Secondary 14E07, 14L30, 14M10.

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The first-named author was partially supported by Simons Foundation Award 546235, and by NSF grant 1701659.

The second-named author was partially supported by NSF grant 2000099.

Received 31 August 2021

Received revised 21 February 2022

Accepted 27 February 2022

Published 25 October 2022