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 $N\_S,N\_T$? 



Non-proper intersection of surfaces
The answer is NO. Take $X=\mathbb{P}^4$, $S,T$ two smooth quadrics. Then $(S\cdot T)=4$, the normal bundles are $N\_S=\mathcal{O}\_S(1)\oplus \mathcal{O}\_S(2)$, and same for $T$. If $S$ and $T$ are general, $Z=0$. If they are given by 
$$S:\ X=0\ ,\ YU+TV=0\quad;\quad T:\ Y=0\ ,\ XV+TU=0$$(coordinates $(X,Y,T,U,V)$ on $\mathbb{P}^4$), then $Z$ is the line $X=Y=T=0$ on $S$, with $Z^2=0$. If $S$ and $T$ lie in the same hyperplane, then $Z$ is a degree 4 elliptic curve in $S$, linearly equivalent to 2 times the hyperplane class $H$, and $Z^2=4H^2=8$.


