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The scheme-theoretic intersection of a projective hypersurface with a projective plane. If the plane is not contained in the hypersurface, restricting the defining homogeneous polynomial gives a projective plane curve of the same degree, with possible repeated components.

At a smooth point of a hypersurface, restriction to the tangent hyperplane has zero linear term. The natural hyperplane intersection therefore has tangent dimension one larger than its expected dimension and is singular at that point.