I want to analyze the ideal $I=(x^{a_1+a_2}-y^{b_1}z^{c_2},y^{b_1+b_2}-x^{a_2 }z^{c_1},z^{c_1+c_2}-x^{a_1}y^{b_2})\subset k[x,y,z]$, where $a_i,b_i,c_i$ are positive integers.
Note $I$ is a determinantal ideal of $A=\begin{bmatrix}x^{a_1}&y^{b_1}&z^{c_1}&x^{a_1}\\ z^{c_2}&x^{a_2} &y^{b_2}&z^{c_2} \end{bmatrix}$.
I am new to determinantal ideals, and I would like to know is there any notable properties of this ideal. It's easy to see that $y^{b_2}(x^{a_1+a_2}-y^{b_1}z^{c_2})\in (y^{b_1+b_2}-x^{a_2 }z^{c_1},z^{c_1+c_2}-x^{a_1}y^{b_2})$, so it would be helpful if $I$ is prime (but seems too good to be true).
Possibly related: Are the determinantal ideals prime?.
