I'm interested in the first basic case of excess intersection in intersection theory:

Let $X$ be a smooth projective 4-fold and let $S,T$ be two surfaces in $X$. Assume that the intersection $S\cap T$ contains an effective 1-cycle $D$ as its 1-dimensional part. In other words, $Z$ defines a Cartier divisor on $S$.

Is there some way of extracting information about $D$ as a divisor on $S$ (e.g., the self-intersection $D^2$) given the intersection number $S\cdot T$ and the normal bundles $N_S,N_T$?

contain$Z$ or itis$Z$? Do you assume some kind of regularity (whatever this may mean in this situation)? $\endgroup$ – Alex Degtyarev Feb 18 '15 at 19:56