Detecting flat normal cones using Segre classes

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Given a flat, projective morphism Y → T from an equidimensional scheme to a nonsingular curve and a subscheme Z of Y, we give conditions under which specialization of the Segre class s(NZY) of the normal cone of Z in Y implies flatness of the normal cone. We apply this result to study when the relative tangent star cone of a flat family is flat.