𝑞-bic forms

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.