Radical ideal computation (Macaulay2)

Question

Is there a way to find the radical ideal of $I_i=(a^n-u^{n-i+1}v^{n-i}, b^n-u^{i-1}v^i, uv-ab)$ for $1\leq i \leq n$ in $\mathbb{C}[u,v,a,b]?$

This is the generalization of my question here where I wanted to use Macaulay2 software to compute the radical ideal for $n=3$ and $i=2$ of the above ideal. Unfortunately, I don't know how to use the software in the general case, and I don't know if it works or not. At least, using Macaulay2 for some special cases I can guess that $\sqrt{I_i}=(a^i-u^{n-i}b^{i-1}, b^{n-i+1}-v^{i-1}a^{n-i}, uv-ab)$ but there is problem in my further computation, so I thought maybe what I guessed is wrong!

I would appreciate any help on that.

Motivation:

This is indeed related to the Derived McKay correspondence where I'm studying the image of torus-invariant, zero-dimensional sheaves of the minimal resolution $Y$ of $\mathbb{C}^2/\mathbb{Z}/n$ under the Fourier-Mukai transform from the (bounded) derived category of coherent sheaves on $Y$ to the (bounded) derived category of coherent sheaves on $\mathfrak{X}=[\mathbb{C}^2/\mathbb{Z}/n],$ the stacky resolution of $\mathbb{C}^2/\mathbb{Z}/n.$

Answer 1

Is this roughly what you want to do?

---

Script: ~/tmp/ideals.m2

R = QQ[u,v,a,b];

n = 10;

myIdeal = (i,n) -> (
    p1 = a^n - (u^(n-i+1))*(v^(n-i));
    p2 = b^n - (u^(i-1))*(v^i);
    p3 = u*v - a*b;
    ideal(p1,p2,p3)
    )

myRad = (I) -> (
    radical I
)

for j from 1 to n do (
    J = myIdeal(j,n);
    rJ = myRad(J);
    print rJ
)

---

Output: (for $1 \leq i \leq 10$)

ii60 : load "~/tmp/ideals.m2"
                 10           9
ideal (- u*v + a*b, b   - v, - u*b  + a)
                     9             8    2
ideal (- u*v + a*b, b  - v*a, - u*b  + a )
                     8      2       7    3
ideal (- u*v + a*b, b  - v*a , - u*b  + a )
                     7      3       6    4
ideal (- u*v + a*b, b  - v*a , - u*b  + a )
                     6      4       5    5
ideal (- u*v + a*b, b  - v*a , - u*b  + a )
                     6      4       5    5
ideal (- u*v + a*b, a  - u*b , - v*a  + b )
                     7      3       6    4
ideal (- u*v + a*b, a  - u*b , - v*a  + b )
                     8      2       7    3
ideal (- u*v + a*b, a  - u*b , - v*a  + b )
                     9             8    2
ideal (- u*v + a*b, a  - u*b, - v*a  + b )
                     10           9
ideal (- u*v + a*b, a   - u, - v*a  + b)

---

Example: ($i=2$, $n=3$)

ii61 : J2 = myIdeal(2,3)

                 2     3       2    3
oo61 = ideal (- u v + a , - u*v  + b , u*v - a*b)

oo61 : Ideal of R

ii62 : myRad(J2)

               2               2
oo62 = ideal (a  - u*b, v*a - b , u*v - a*b)

oo62 : Ideal of R