I need to compute $\dim Z(x_1x_2,x_2x_3,x_1x_3)$, in $\mathbb{A}_k^3$. I believe that this variety consists of the unique point $(0,0,0)$. So, its dimension is $0$. Am I correct? Many thanks in advance!
2026-03-25 20:41:56.1774471316
$\dim Z(x_1x_2,x_2x_3,x_1x_3)=0$?
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Collating what was in the comments, $X=V(xz,xy,yz)$ is the union of the coordinate axis, so has an isolated singularity at the origin. The tangent space at any non-singular point is therefore a line, which is one-dimensional, hence $\text{dim}\,X=1$.