How can I get maximal ideal containing an ideal using Macaulay2?

Question

In Macaulay2, I have written the following codes to find the maximal ideal in the ring $Q[x,y,z]$ containing the ideal generated by $x^2y+z$ and $xz-y$.

R=QQ[x,y,z]
I=ideal(x^2y+z,xz-y)
M=getMaxIdeal I

But every time I get the error code:

   error: no method for adjacent objects:<br> 

getMaxIdeal (of class Symbol)
-- 2 -- SPACE ideal (x y + z, x*z - y) (of class Ideal)

Please help me.

Answer 1

The scheme $X\subseteq {\Bbb A}^3$ defined by the ideal $I=\langle x^2y+z,zx-y\rangle$ is supported

R=QQ[x,y,z]
I=ideal(x^2*y+z,z*x-y)
minimalPrimes I -- {ideal(z,y), ideal(x+1,y+z), ideal(x^2-x+1,-x*z+y)}

on the lines $y=z=0, x+1=y+z=0,$ and the lines $x^2-x+1=xz-y=0.$

The lines intersect as as follows:

minimalPrimes(ideal(z,y)+ideal(x+1,y+z)) -- {ideal(z,y,x+1)}
minimalPrimes(ideal(z,y)+ideal(x^2-x+1,-x*z+y)) -- {ideal(x^2-x+1,z,y)}
minimalPrimes(ideal(x+1,y+z)+ideal(x^2-x+1,-x*z+y)) -- {}

So $X$ has singular locus the three (1+2) points: {ideal(z,y,x+1), ideal(z,y,x^2-x+1)}. These give rise to the distinguished maximal ideals or points.

If you want any other points on the lines, just pick points on, say ideal(z,y), by picking a third plane, say $x-a=0$, giving the maximal ideal, say ideal(z,y,x-a) for any $a$ in your field.