I'm attempting to use Macaulay2 to compute minimal prime decompositions of various ideals. I'm getting used to the program and I've been successful in simpler cases. But with the ideal I present below, Macaulay2 gets completely stuck, even after letting it run a long time, as well as trying the computation on a much more powerful machine. I'm wondering if there is some theoretical reason Macaulay2 cannot handle it, or if there is some round-about way I can get the minimal primes.
Anyways, I let R=QQ[z] and I=ideal(z^2-z+1) and work in the polynomial ring S=R/I[x0,x1,x2,x3,x4,x5,y0,y1,y2,y3,y4,y5]. I define the ideal of S:
J=ideal(x0*x3 - x1*x4, x0*x3 - x2*x5, x1*x4 - x2*x5, y0*y3 - y1*y4,
y0*y3 - y2*y5, y1*y4 - y2*y5, ((x0+x1)*(x0+x2)*(x1+x2)-8*x0*x1*x2)*(x0*(x1^2-x2^2)+(z^2)*x1*(x2^2-x0^2)+(z^4)*x2*(x0^2-x1^2))*y0-(x1-x2)*(x2-x0)*(x0-x1)*(x0+(z^4)*x1+(z^2)*x2)*(x1*x2+(z^4)*x0*x2+(z^2)*x0*x1)*y1, (x0*(x1^2-x2^2)+(z^4)*x1*(x2^2-x0^2)+(z^2)*x2*(x0^2-x1^2))*(x0+(z^4)*x1+(z^2)*x2)*(x1*x2+(z^4)*x0*x2+(z^2)*x0*x1)*y1-(x0*(x1^2-x2^2)+(z^2)*x1*(x2^2-x0^2)+(z^4)*x2*(x0^2-x1^2))*(x0+(z^2)*x1+(z^4)*x2)*(x1*x2+(z^2)*x0*x2+(z^4)*x0*x1)*y2, (x1-x2)*(x2-x0)*(x0-x1)*(x0+(z^2)*x1+(z^4)*x2)*(x1*x2+(z^2)*x0*x2+(z^4)*x0*x1)*y2-((x0+x1)*(x0+x2)*(x1+x2)-8*x0*x1*x2)*(x0*(x1^2-x2^2)+(z^4)*x1*(x2^2-x0^2)+(z^2)*x2*(x0^2-x1^2))*y0);
The goal is to find the minimal prime decomposition by computing minimalPrimes J. But Macaulay2 just works indefinitely without success here. As I said above, even on a more powerful machine in my department. Relatively speaking, this is a pretty simple ideal, so I'm not sure what's going on.
The computation does work if you add another function to J, so is the problem that somehow this variety is underdetermined or something? Geometrically, this is the total transform of a subvariety under a blow up. And I need to use minimalPrimes to find the proper transform.