How independent are the minors of a matrix? Let $M$ be a $p\times q$ matrix (say, $p\leq q$) with formal coefficients $m_{i,j}$. It has $pq$ entries, and ${q\choose p}$ minors of maximal size $p\times p$, which gives us a map $$\varphi_{p,q}: \Bbb Z^{pq}\to \Bbb Z^{q\choose p}$$
I'm interested in the surjectivity of this map. It is easy to see that $\varphi_{3,4}$, for instance, is surjective, meaning that we can adjust the coefficients of a $3\times 4$ matrix to give each of the $3\times 3$ minors a prescribed value.
In most cases, ${q\choose p}$ is much bigger than $pq$, so it feels unlikely that $\varphi$ will be surjective. But what if we keep only a relatively small amount of minors?

Question 1: For what values of $p,q,r\in\Bbb N$ is there a map $f:\Bbb Z^{q\choose p}\to \Bbb Z^r$ defined by keeping only some $r$ coordinates, such that $f\circ \varphi_{p,q}$ is surjective?
Question 2 (less general but the one I'm really interested in) : Take $p=3, q=6$ and forget about the three minors corresponding to columns $(1, 4, 5), (2, 4, 6), (3, 5, 6)$. Is the resulting map $\Bbb Z^{18}\to \Bbb Z^{17}$ surjective?

 A: Thanks to a comment by @ancientmathematician, I was able to focus my research and to answer Question 2: the $\Bbb Z^{18}\to \Bbb Z^{17}$ map from the question is not surjective. However, the way to the answer yields new interesting questions. First, here is the proof.

Theorem: an ordered collection of $q\choose p$ integers is the collection of (lexicographically ordered) maximal minors of some
$p\times q$ matrix if and only if these numbers satisfy a set of polynomial equations known as
Plücker equations.


Context: see these lecture notes by Alexander Yong.
Proof: see Schubert Calculus by Kleiman and Laksov.

Plücker equations for $(p,q)=(3,6)$ can be displayed by typing Grassmannian(2,5) into Macaulay2 (the $2$ and $5$ come from projective reasons).
Here is one of these equations:
$$p_{2,3,4} p_{1,3,6} -p_{1,3,4}p_{2,3,6} +p_{1,2,3}p_{3,4,6}=0$$
This one, and five others, involve only determinants that I wanted to keep (none of $p_{1,4,5}, p_{2,4,6}, p_{3,5,6}$ is involved). Hence the result: the $\Bbb Z^{18}\to \Bbb Z^{17}$ map from the question is not surjective.

New problem as promised: what if we forget about enough minors that each of the Plücker equations involves at least one of them (basically making the argument above fail)? We can't conclude right away, but what tools would one use to conclude then? I have been thinking of using Groebner bases, but it doesn't seem straightforward.
