  A frame template over a field $\mathbb F$ describes the precise way in which
a given $\mathbb F$-representable matroid is close to being a frame matroid.
Our main result determines the maximum-rank projective or affine geometry that
is described by a given frame template over a prime field. Subject to the
matroid minors hypothesis of Geelen, Gerards, and Whittle, we use our result to
determine, for each projective or affine geometry $N$ over a prime field
$\mathbb F$, a best-possible upper bound on the number of elements in a simple
$\mathbb F$-representable matroid $M$ of sufficiently large rank with no
$N$-minor.
