FUNCTION-FIELDS OF GENERALIZED BRAUER-SEVERI VARIETIES

We define a generalized Brauer-Severi variety to be the projective variety whose closed points correspond to rank m right ideals in a central simple algebra. We show that these varieties are forms of the Grassmann variety and their function fields are generic partial splitting fields for the associated algebras. The subgroup of the Brauer group split by the function field is calculated. Following Schofield and van den Bergh, we calculate the index of a simple algebra extended by the function field of a generalized Brauer-Severi variety associated with any other central simple algebra.