Let $X$ be a projective complex manifold. Consider two different irreducible hypersurfaces $Y,Z\subset X$, with cohomology classes $[Y],[Z]\in H^2(X)$. Normally, one would expect that $[Y\cap Z]=[Y]\smile[Z]\in H^4(X)$. But, as I understand it, this is not always the case. A simple example I know is $X=V(xy+zw+uv)\subset P^5$, $Y=X\cap V(u)$, $Z=X\cap V(z)$. In this case the intersection $Y\cap Z$ has two irreducible components and, no matter how you look at it, its cohomology class is not $[Y]\smile[Z]$.

Question What is a correct formulation of the rule $[Y\cap Z]=[Y]\smile[Z]$ for hypersurfaces?

(I tried to find this in Fulton, but failed. Admittedly, this does not tell very much, as my native subject is differential geometry.)