Quotients of tangential $k$-blocks

A tangential k-block over GF(q) is a simple matroid representable over GF(q) with critical exponent k + 1 for which every proper loopless minor has critical exponent at most k. Such matroids are of central importance in the critical problem of Crapo and Rota. In this paper we provide sufficient conditions for a quotient of a tangential k-block over GF(q) to be also a tangential k-block over GF(q). This enables us to show that there exist rank r supersolvable tangential k-blocks over GF(q) exactly when qk > r > k + 1.