Is it correct that we can write the affine variety $V(xy, xz, yz)$ in $\mathbb{R}^3$ as $$[V(x) \cup V(y, z)] \cup [V(x, y) \cup V(z)]?$$ Is there another way to write this in order to better imagine the variety?
2026-03-26 12:38:27.1774528707
Simpler way to express the affine variety V(f, g, h) in terms of union and intersection of varieties
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Just use the following
$$\begin{align} &V(f,g,h)=V(f)\cap V(g)\cap V(h)\\ &V(f\cdot g)=V(f)\cup V(g) \end{align}$$
One has
$$ \begin{align} V(xy,xz,yz)&=V(xy)\cap V(xz)\cap V(yz)\\ &=\left(V(x)\cup V(y)\right)\cap\left(V(x)\cup V(z)\right)\cap\left(V(y)\cup V(z)\right) \end{align} $$
Can you take it from here using
$$A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$$