Quadric surface bundles over surfaces

Abstract

Let $f : T \to S$ be a finite flat morphism of degree 2 between regular integral schemes of dimension $<= 2$ with 2 invertible, having regular branch divisor $D \subset S$. We establish a bijection between Azumaya quaternion algebras on $T$ and quadric surface bundles with simple degeneration along $D$. This is a manifestation of the exceptional isomorphism $^2\mathsf{A}_1=\mathsf{D}_2$ degenerating to the exceptional isomorphism $\mathsf{A}_1=\mathsf{B}_1$. In one direction, the even Clifford algebra yields the map. In the other direction, we show that the classical algebra norm functor can be uniquely extended over the discriminant divisor. Along the way, we study the orthogonal group schemes, which are smooth yet nonreductive, of quadratic forms with simple degeneration. Finally, we provide two applications: constructing counter-examples to the local-global principle for isotropy, with respect to discrete valuations, of quadratic forms over surfaces; and a new proof of the global Torelli theorem for very general cubic fourfolds containing a plane.