Bounds for minimal sized frames which are injective for phase retrieval (a problem in signal processing) have been obtained using the fact that the variety of $m \times n$ matrices of rank at most $k$ is irreducible of codimension $(m - k)(n - k)$ in the projective space of nonzero $m \times n$ matrices (specifically in the case $n=m$ and $k=2$). See the paper of Balan, Casazza and Edidin or Conca, Edidin, Hering, and Vinzant for details.