Let $M$ be a smooth manifold of dimension $m$ and $\pi:E\rightarrow M$ a vector bundle of rank $e$. Given a section $s$ of the bundle $\pi:E\rightarrow M$, we expect that the zero locus $Z(s)$ of $s$ is a submanifold $N\subset M$ of dimension $m-e$. This is true if $s$ can be perturbed into a general position so that $s(M)$ and the zero section intersect transversally.
Perturbation is not always possible (for example in holomorphic category category). In this case we need "excess intersection theory"; if the section $s$ lies in a subbundle $F\subset E$ and is a transversal section of $F$, the correct $(m-e)$-cycle we should take is the Euler class of the quotient bundle $E/F$, which is homologous to $Z(s)$ if transverse perturbation of $s$ exists.
My problem is that I don't really know good explicit examples with which I can compute things. Could anyone give me an example or reference, which shows how useful excess intersection theory is?
Edit My motivation to study excess intersection theory is virtual cycles of moduli spaces, which of course are very good examples of excess intersection theory. But I am looking for some elementary examples on which I can compute things. I want to convince myself that the theory is really reasonable by computing a few simple examples.