#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), 31-70

** Asher Auel, R. Parimala, and V. Suresh **
Quadric Surface Bundles over Surfaces

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 ${}^2\Dynkin{A}_1=\Dynkin{D}_2$
degenerating to the exceptional isomorphism $\Dynkin{A}_1=\Dynkin{B}_1$.
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.

2010 Mathematics Subject Classification: 11E08, 11E20, 11E88, 14C30, 14D06, 14F22, 14L35, 15A66, 16H05

Keywords and Phrases: quadratic form, quadric bundle, Clifford algebra, Azumaya algebra, Brauer
group, orthogonal group, local-global principle, cubic fourfold, K3 surface

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