I have a reducible variety, $X = \bigcup_{i \in I} X_i \subset Y$, and I want to understand the normal cone $C_X Y$.

I write $X_J$ for $\cap_{j \in J} X_j$.

Can $C_X$ be expressed in terms of the $C_{X_J} X_K$?

(I take the convention $X_\emptyset = Y$ so that the $C_{X_i} Y$ are among the above.)

In fact in the example that has prompted this question, all the $X_J$ are smooth and connected, (though I do not know that they intersect transversely), and I am actually only interested in numerical information; i.e.,

Can the Segre class $S(X,Y)$ be expressed in terms of the Segre classes $S(X_J, X_K)$ for $J \subset K$ and perhaps also the intersection multiplicities of $X_M$ and $X_N$ along their intersection?