Show that $Z(XY,XZ,YZ)$ is not irreducible.
what I think it is $Z(XY,XZ,YZ)=Z(XY)∩Z(XZ)∩Z(YZ)=(Z(X)∪Z(Y))∩(Z(X)∪Z(Z))∩(Z(Y)∪Z(Z))$
Then I am not sure how to carry on, and what I need to show it is $Z(XY,XZ,YZ)$ is a union of two proper subvarieties.
Can anybody help please, thanks a lot.
The answer is that this is equal to the union of the three coordinate axes: $$Z(X, Y) \cup Z(X,Z) \cup Z(Y,Z)$$
Look at the three polynomials $XY, XZ, YZ$. Whenever you have a point on one of the three axes, at least two of the three coordinates equal $0$, which means that all three polynomials evaluate to $0$. If you have a point that is not on any of the axes, then there are at least two non-zero coordinates, and the corresponding polynomial therefore does not evaluate to $0$.