# Intersection inside normal cone

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 ?