Some modular varieties of low dimension II

Some years ago one of the two authors in collaboration with F. Hermann studied in [FH] some modular varieties of small dimension. These are related to the orthogonal group O(2, n). In particular using some exceptional isogenies between orhogonal and symplectic groups, they used techniques of both “the worlds”. The most significant variety they studied in [FH] was related to O(2, 6) or —equivalently— to the symplectic group of degree two defined on the quaternions. In particular they gave a finite map from this variety to a weighted 6-dimensional projective space. Recently Krieg determined the structure of the graded ring of modular forms of degree two with respect to the Hurwitz integral quaternions. Krieg’s result convinced the authors to reconsider [FH]. Here we found that in the computation of the covering degree of the mentioned finite map a factor three is missing in the denominator. The aim of this paper is to correct this mistake and to improve the result of [FH]. In this way we will get Krieg’s structure theorem as a corollary of a more general result. In contrast to Krieg we work only in the orthogonal context. We think that this makes the underlying geometry much more visible.