$q$-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.

Comments very welcome! v2: minor edits; v3: update references and introduction

References in corpus (2)

• $q$-bic hypersurfaces and their Fano schemes
• Free curves in Fano hypersurfaces must have high degree

Cited by in corpus (1)

• $q$-bic threefolds and their surface of lines