I've been studying algebraic geometry recently and there is a problem I'm struggling with:
Suppose $A$ is a $m\times n$ complex matrix of rank $\leq r$, this is equivalent to all its $(r+1)\times (r+1)$ minors vanish. Hence the determinants of these minors define a projective variety. Then consider its ideal. I wonder whether these minors form a reduced Grobner basis for such $A$.
Theorem 1 in this paper seems to prove this but for generic matrices.
What does "generic" mean in this context? Can we just say the minors form a reduced Grobner basis for any complex matrix with rank $\leq r$?