Representability of matroids with a large projective geometry minor

We prove that, for each prime power q, there is an integer n such that if M   is a 3-connected, representable matroid with a PG(n−1,q)PG(n−1,q)-minor and no U2,q2+1U2,q2+1-minor, then M   is representable over GF(q)GF(q). We also show that for ℓ≥2ℓ≥2, if M   is a 3-connected, representable matroid of sufficiently high rank with no U2,ℓ+2U2,ℓ+2-minor and |E(M)|≥2ℓr(M)/2|E(M)|≥2ℓr(M)/2, then M is representable over a field of order at most ℓ.