I am studying Intersection Theory in algebraic geometry and in the following when I say intersection I mean the intersection of a dimension k cycle with a codimension k cycle.
I was reading this question (and in particular its answers) in order to clear my mind in what does it mean that the intersection of two cycles is negative (the question I have linked only refers to the case of two curves in a surface).
But I am interested in the higher dimensional case. Let's consider the case of a curve $C$ intersects with a divisor $D$ (codimension 1). For what I can understand, saying that $C\cdot D<0$ in particular means that the curve $C$ is contained in $D$. Is it true? Is it the definition in some sense?
Moreover, which is the geometrical information contained in $C\cdot D=-k$ for $k>0$ (I am thinking in the case in which $C\cdot D=-2$ for example).
Thank you very much for any hint, advice and suggestion. And of course references will be very welcome (I was trying to read Eisenbud and Harris '3264 & Intersection Theory', but I cannot realise this situation...)!
EDIT: I was wondering also about the following. Somewhere I found that a 'simple' k-cycle is a reduced and irreducible subvariety of (co)dimesion k. Somewhere else I instead found that a k-cycle is only irreducible. So it may be that when one writes $C\cdot D=-k$ he geometrically means that $C\subset D$ and that $C$ is not reduced and its local ring has length $k$. Can it be right?