Matroids denser than a projective geometry

The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each non-negative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$. The Growth-Rate Theorem of Geelen, Kabell, Kung, and Whittle shows, essentially, that the growth-rate function is always either linear, quadratic, exponential with some prime power $q$ as the base, or infinite. Morover, if the growth-rate function is exponential with base $q$, then the class contains all GF$(q)$-representable matroids, and so $h(r)\ge \frac{q^r-1}{q-1}$ for each $r$. We characterise the classes that satisfy $h(r) = \frac{q^r-1}{q-1}$ for all sufficiently large $r$. As a consequence, we determine the eventual value of the growth rate function for most classes defined by excluding lines, free spikes and/or free swirls.