  Scheme-theoretic methods are used to classify ternary quadratic forms with
values in line bundles over arbitrary schemes and to canonically determine the
isomorphisms between them. The association of a quadratic bundle to its even
Clifford algebra induces a natural bijection from the set of equivalence
classes of line-bundle-valued quadratic forms on rank 3 vector bundles upto
tensoring by twisted discriminant bundles and the set of isomorphism classes of
schematic specialisations of rank 4 Azumaya bundles over any fixed scheme X.
This statement is a limiting version of the following statement: the set of
orbits of Disc(X) in the 1-cohomology of X in the fppf topology with values in
O(3) is in bijection with the 1-cohomology with values in PGL(2). The various
orthogonal groups of a quadratic bundle are canonically determined in terms of
the automorphisms of its even Clifford algebra. Any automorphism of the latter
arises from a similarity, and in fact from an orthogonal transformation if its
determinant is a square. The special orthogonal group is thus identified with
the subgroup of automorphisms with trivial determinant. A specialised algebra
arises from a honest quadratic form iff its determinant has a square root and
arises from a bilinear form iff the line subbundle generated by 1 is a direct
summand.
