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For a regular embedding $X\subset Y$, one can think the intersection class $[X]\cdot [X]$ as the intersection of the perturbation of the zero section inside $N_X Y$, intersect with itself. For non-regular embedding $X\subset Y$, with $Y$ smooth, one has to follow Fulton's construction, to intersect the normal cone $C_X X\times X$ with the zero section inside $TY|_X$.

My question is, why can't w take a perturbation of the (real) zero section of $X\to C_X Y$, and count the intersection as $[X]\cdot [X]$? Are there any known counter examples ?

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