first a disclaimer: I am not an expert in alg. geometry so please don't shoot. Suppose X is a closed subscheme (not nec. reduced, and $dim >0$) of a smooth (projective if you want) variety Y. Suppose (for simplicity) that X is pure dimensional and Z is an irreducible component of X. The multiplicity of Y along X at Z, denoted $(e_XY)_Z$ can be computed either via the Hilbert-Samuel polynomial or Segre classes of the normal cone (Fulton Example 4.3.4). This seems to be a global description of the multiplicity. My question is whether one can localize and compute this quantity at a generic point of Z? A related question: the fibers of the normal cone $C_XY$ are generically isomorphic along Z? Any (newer) reference for a more indepth study of $C_XY$?
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