2005/06/08 by Venkata Balaji Thiruvalloor Eesanaipaadi, Eesanaipaadi, Venkata Balaji Thiruvalloor
Mathematics · #14A25 #14F05 #14L15 #14M #14Q #15A63 #15A66 #15A75 #15A78 #16H05 #16S60 #16W20 #20G05 #20G35 #Algebraic Geometry (math.AG) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AG #math.RA #msc:14A25 #msc:14F05 #msc:14L15 #msc:14M #msc:14Q #msc:15A63 #msc:15A66 #msc:15A75 #msc:15A78 #msc:16H05 #msc:16S60 #msc:16W20 #msc:20G05 #msc:20G35
paper · pdf · doi:10.48550/arxiv.math/0506146
79 pages; PDFLaTeX; improved exposition; new result included--isomorphism of the Picard group of the base with that of the scheme of specialised algebras; some typos and cross-references corrected
arxiv created 2005/07/24 · arxiv updated 2009/12/01
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.