I'm trying to understand why the intersection multiplicity of two singular subvarieties is not equal to the complex dimension of the local ring but it is instead the Euler characteristic. Is it possible to find an intuitive explanation? I think that the following concepts need some clarification: 1) when do smooth curves meet transversally? 2) why have we to focus on the group of invertible sheaves? 3) any simple examples of pull-back of line bundles?