https://drive.google.com/file/d/1MzElOy1tgJGzZkCpzY8d_msKHuV-03kx/view?usp=sharing
Here is a link to Strumfels 1990 paper(Grobner Basis and stanley decomposition of determinantal rings (pdf pg-137)).
Theorem $1$ in this paper is stated as follows:
The set of $(r+1 \times r+1) $ minors of the matrix forms a Grobner basis of $I_r$ with respect to the lexicographical ordering $x_{1,n} > x_{1,n-1} > \cdots x_{m,1}$ where $I_r$ denotes the vanishing sub-ideals of the affine subvarieties of matrices of rank at most $r < min\{m,n\}$.
I don't understand the term vanishing sub-ideals of the affine subvarieties of matrices of rank at most $r < min\{m,n\}$ .
My doubt also lies in the fact does it imply somehow that the set of all $k \times k $ minors of a generic $n \times n$ matrix forms a grobner basis of the ideal $I_k(X)$ where $I_k(X)$ denotes the ideal generated by $k \times k$ minors of the matrix $X$ of the form $[a_1 \cdots a_k \mid 1 \cdots k]$ where $1 \cdots k$ are columns and $a_1 \cdots a_k$ are rows.
Edit 1: (i)$C_k:= \{a = (a1, ··· , ak) |1 ≤a1< ··· <ak≤n\}$ denotes the collection of all ordered k-tuples from $\{1, ··· , n\}. $
(ii) Given $ a = (a_1, ..., a_k)∈C_k$;
$X^a= [a_1, ··· , a_k|1, 2, ..., k]$denotes the $k ×k $ minor of the matrix $X$, with $a_1, ..., a_k$ as rows and $1, ..., k$ as columns.
•$S_k:= \{X^a:a ∈C_k\}$and $I_k$ denotes the ideal generated by $S_k$ in the polynomial ring R
Lemma 4.2 The Groebner Basis of the ideal $I_k$ with respect to the monomial ordering is the set $S_k$.
The proof uses Buchberger's criterion
$S(X^c,X^d) \to_{(S_k)} r$. Then the set $S(X^c,X^d) - \sum_{a_i \in C_i}(h_i.X^{a_i}) = r$.
If $X$ is generic, then the set of all $k \times k$ monomial ordering forms a Groebner basis with respect to the the chosen monomial ordering.(strumfels 1990)(**)
Therefore there exists $[a_1,a_2,...,a_k \mid b_1 \cdots b_k]$ such that $\prod_{i=1}^k(\chi_{a_i.b_i})$ divides the $LT(r)$. We see that if $b_k=k$ then we are done.
Let $X$ be generic symmetric .Then, $a_k=k$ and $b_k \ge k+1$ imply that the minor belongs to the set $S_k$.(Can someone explain this property of generic symmetric matrix?)
Let $X$ be generic. Then, for any $a_k$ and under the condition $b_k \ge k+1$, then $\chi_{a_k.b_k} | LT(r)$ but $\chi_{a_k.b_k}$ doesn't divide any term of the elements in $S_k$. (Also I need some help in this property of generic symmetric matrix?)
I dont understand the (**) as to where it is mentioned in the paper .
Edit 2:I was going through the proof and I got stuck in the two properties of the generic matrices.Can someone explain them to me ?
He's just freely using some well-known facts. First, the matrices with rank at most $r < \min\{m, n\}$ are those for which all $(r+1) \times (r+1)$ minors vanish. Second, the ideal generated by these minors is a radical ideal, so it's precisely equal to the vanishing ideal of the set of matrices with rank at most $r$. Hence the quotient is exactly the coordinate ring of that set of matrices, which is an affine algebraic subvariety of the full affine algebraic variety of matrices. (He could perhaps say "ideal" instead of "subideal", though that's a matter of taste.)
I don't know why you're suggesting you can restrict the column indices to $1, \ldots, k$. I don't think there's any hope that anything like that is true.
See this MO thread for related discussion and further details. (Beware minor notational mismatches.)