Line-Bundle-Valued Ternary Quadratic Bundles Over Schemes

Abstract

We describe a satisfactory theory of degeneration of quadratic forms in three variables in the most general setting possible: the quadratic forms are defined on rank 3 vector bundles over an arbitrary scheme and could have values in nontrivial line bundles. Our results extend what is known for good forms; for example we show that the Witt-invariant suffices for classification. We determine explicitly the general, special and usual orthogonal groups and present applications. We indicate examples of rank 4 vector bundles that do not admit any Azumaya structures and of rank 3 vector bundles that do not admit any good quadratic forms with values in specified line bundles. These examples occur naturally on the Seshadri-desingularisations of moduli spaces of rank two degree zero vector bundles over a curve relative to an integral normal locally-Nagata (universally Japanese) base scheme.