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.