2012/07/17 by Asher Auel, Auel, Asher, R. Parimala +3
Mathematics · #11E08 #11E20 #11E88 #14D06 #14F22 #14L35 #15A66 #16H05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1207.4105
openalex publication_date 2012/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let T -> S be a finite flat morphism of degree two between regular integral schemes of dimension at most two (and with 2 invertible), having regular branch divisor D. We establish a bijection between Azumaya quaternion algebras on T and quadric surface bundles over S with simple degeneration along D. This is a manifestation of the exceptional isomorphism A12 = D2 degenerating to the exceptional isomorphism A1 = B1. 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 surprising 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.