Fulton and Macpherson suggests the way to compactify the set of $n$-labelled distinct point on variety in their paper, "A Compactification of Configuration Spaces"
In this paper, the process and the motivation is very simple, and hence the construction itself is not that hard to understand.
However, I couldn't check the motivation of $X[n]^+$. In the paper, it is for describing $n$-th degenerated configuration of $X$. In what sense, the fiber of $X[n]^+ \to X[n]$ becomes such a configuration? I tried to understand this by letting $X = \mathbb{P}^1$, but still don't know unified description of whole phenomenon.
