Rationality of cycles modulo 2 on products of generically smooth quadrics in characteristic 2

TL;DR

This paper extends the algebraic-geometric approach to analyze rational cycles modulo 2 on products of generically smooth quadrics in characteristic 2, aiding the study of quadratic forms and their correspondences.

Contribution

It broadens the study of rational cycles modulo 2 on products of quadrics in characteristic 2, enabling analysis of correspondences between smooth and non-smooth quadrics.

Findings

Extended tools for rational cycles modulo 2 on products of quadrics in characteristic 2.

Applications to degenerate quadratic forms in characteristic 2.

Discussion of open problems in quadratic form theory.

Abstract

A 2022 result of Karpenko establishes a conjecture of Hoffmann-Totaro on the possible values of the first higher isotropy index of an arbitrary anisotropic quadratic form of given dimension over an arbitrary field. For nondegenerate forms, this essentially goes back to a 2003 article of the same author on quadratic forms over fields of characteristic not $2$. To handle the more involved case of degenerate forms in characteristic $2$, Karpenko showed that certain aspects of the algebraic-geometric approach to nondegenerate quadratic forms developed by Karpenko, Merkurjev, Rost, Vishik and others can be adapted to a study of rational cycles modulo $2$ on powers of a given generically smooth quadric. In this paper, we extend this to a broader study of rational cycles modulo $2$ on arbitrary products of generically smooth quadrics in characteristic $2$. A basic objective is to have tools available to study correspondences between general quadrics, in particular, between smooth and non-smooth quadrics. Applications of the theory to the study of degenerate quadratic forms in characteristic $2$ are provided, and a number of open problems on forms of this type are also formulated and discussed.