$q$-bic forms

TL;DR

This paper develops a geometric classification framework for $q$-bic forms, introducing filtrations and invariants to analyze their automorphisms and parameter space relations.

Contribution

It introduces a geometric theory of $q$-bic forms, including filtrations and invariants, to classify and study their automorphism groups and parameter space relations.

Findings

Two intrinsic filtrations associated with $q$-bic forms

Numerical invariants derived from these filtrations

Classification and automorphism analysis of $q$-bic forms

Abstract

A $q$-bic form is a pairing $V \times V \to \mathbf{k}$ that is linear in the second variable and $q$-power Frobenius linear in the first; here, $V$ is a vector space over a field $\mathbf{k}$ containing the finite field on $q^2$ elements. This article develops a geometric theory of $q$-bic forms in the spirit of that of bilinear forms. I find two filtrations intrinsically attached to a $q$-bic form, with which I define a series of numerical invariants. These are used to classify, study automorphism group schemes of, and describe specialization relations in the parameter space of $q$-bic forms.