  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.
