Let $R=k[x,y,z]$ be a polynomial ring and $f,g,h\in k[x,y,z]_d$ be degree $d$ elements such that $f,g,h$ are homogeneous polynomials of degree $d$ and form a regular sequence. Find the necessary and sufficient condition for $$(x^{2d-1},x^{2d-2}y,\dots,y^{2d-1})=(x,y)^{2d-1}\subset (f,g,h)$$ in terms of the generators $f,g,h$.
The hint is to use Koszul complex, and I know since $f,g,h$ is a regular sequence the Koszul complex gives a minimal free resolution of $R/(f,g,h)$. But I don't see how resolution plays role in this problem. This question rises when I study determinantal ideals and determinantal varieties, a possible reference is 'Algebraic Geometry' by Harris.
Update: I am able to find the minimal resolutions of $R/(f,g,h)$ and $R/(x,y)^{2d-1}$ if they are correct:
$0\to R[-3d]\to R[-2d]^3\to R[-d]^3\to R\to R/(f,g,h)\to 0$
$0\to R[-2d]^{2d-1}\stackrel{\phi}\to R[-2d+1]^{2d}\to R\to R/(x,y)^{2d-1}\to 0$
Here $\phi$ is a $2d\times (2d-1)$ matrix with entries $a_{i,i}=-y$, $a_{j+1,j}=x$ for every $i,j$, and $0$ for other entries.
So what does $(x,y)^{2d-1}\subset (f,g,h)$ tell us about the relationships of these minimal free resolutions?