  Let $s,n \ge 2$ be integers. We give a qualitative structural description of
every matroid $M$ that is spanned by a frame matroid of a complete graph and
has no $U_{s,2s}$-minor and no rank-$n$ projective geometry minor, showing that
every such matroid is `close' to a frame matroid. We also give a similar
description of every matroid $M$ with a spanning projective geometry over a
field GF$(q)$ as a restriction and with no $U_{s,2s}$-minor and no
PG$(n,q')$-minor for any $q' > q$, showing that such an $M$ is `close' to a
GF$(q)$-representable matroid.
