  Traditional algebraic geometric invariants lose some of their potency in
positive characteristic. For instance, smooth projective hypersurfaces may be
covered by lines despite being of arbitrarily high degree. The purpose of this
dissertation is to define a class of hypersurfaces that exhibits such
classically unexpected properties, and to offer a perspective with which to
conceptualize such phenomena.
  Specifically, this dissertation proposes an analogy between the eponymous
$q$-bic hypersurfaces -- special hypersurfaces of degree $q+1$, with $q$ any
power of the ground field characteristic, a familiar example given by the
corresponding Fermat hypersurface -- and low degree hypersurfaces, especially
quadrics and cubics. This analogy is substantiated by concrete results such as:
$q$-bic hypersurfaces are moduli spaces of isotropic vectors for a bilinear
form; the Fano schemes of linear spaces contained in a smooth $q$-bic
hypersurface are smooth, irreducible, and carry structures similar to
orthogonal Grassmannian; and the intermediate Jacobian of a $q$-bic threefold
is purely inseparably isogenous to the Albanese variety of its smooth Fano
surface of lines.
