= Embedding dimension distinguishes three-branch curves
Three coordinate axes in affine three-space have coordinate ideal $(xy,xz,yz)$; their common point has <embedding dimension> three. Three distinct coplanar lines meeting at one point have embedding dimension two there. An isomorphism preserves the unique point lying on all three irreducible components and preserves the <algebraic cotangent space>, so these two configurations are not isomorphic despite identical component incidence.
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