A Cancellation Theorem for Segre Classes

In [1, 4], Fulton defines the notion of the Segre class s(X, Y) ∈ A∗X of a closed embedding of schemes X→ Y over a field k. As in Fulton, all of our schemes are finite type over a ground field k which may be of arbitrary characteristic unless stated otherwise. The Segre class allows us to measure the way in which X sits inside Y, and is functorial for sufficiently nice maps ([1, 4.2]). One important case is the embedding of a closed point, for which the Segre class gives us its multiplicity. Suppose we have an embedding X → Y and the schemes in question embed into a simpler space Z such that Y → Z is a regular embedding. For example, Y could be an intersection of hyper-surfaces in Z = Pk . In this setup, it is natural to ask whether we can calculate s(X, Y), assuming that we understand s(X,Z) and c(NYZ) where NYZ is the normal bundle of the regular embedding Y → Z. In other words, can we deduce intersection theoretic invariants of the possibly complicated embedding X→ Y, from the hopefully simpler embeddings into Z? Fulton provides the answer to this question in one very special case: