Modularity of tangential k-blocks

This paper examines the role of modularity in tangential k-blocks over GF(q). It is shown that if M is a tangential k-block over GF(q) and F is a modular flat of M which is affine over GF(q) then the simple matroid associated with the complete Brown truncation of M by F is also a tangential k-block over GF(q). This enables us to construct tangetial k-blocks over GF(q) of all ranks r where q^k − q + 2⩽r ⩽ q^k. We also consider tangential k-blocks which have modular hyperplanes; bounds are placed on the rank of members of this class and some of their minors are exhibited.