q ‐Lifts of Tangential k ‐Blocks

A tangential k ‐block over GF ( q ) is a geometry representable over GF ( q ) with critical exponent k +1 for which every proper loopless minor has critical exponent at most k . We define what is meant by a q ‐lift of a matroid representable over GF( q ) and show that, if M is a tangential k ‐block over GF( q ) and M ′ is a q ‐lift of M , then M ′ is a tangential k + 1‐block over GF( q ). This enables us to extend the class of known tangential k ‐blocks and to answer some natural questions concerning the existence of modular hyperplanes in tangential k ‐blocks.