Algebraic Geometry of Ternary Quadratic Forms and Orders in Quaternion Algebras

摘要

The form f (a ternary quadratic with integral coefficients) defines two objects, a Z-scheme M(f) and a Z-order O(f), where Z denotes the integers. If f = X squared + Y squared + Z squared, then M(f) = Proj (Z(x,y,z)/(f)) and O(f) is the ring of the Lipschitz integral quaternions. Relations between these two kinds of object are sought. Results concerning models of extensions E/A are reviewed, where A is a perfect Dedekind ring and E a regular extension of genus O of the field of fractions of A and their connections with orders in quaternion algebras. If M and M prime are two relatively minimal models of E/A, then there is a sequence of relatively minimal models M sub O = M, M sub 1, ..., M sub n = M prime such that M sub i + 1 is an elementary transform of M sub i for O or equal i n.