The Wikipedia entry on intersection theory contains the following statement:
[for C a curve, on a surface] "the self-intersection points of C is the generic point of C, taken with multiplicity C · C."
This statement is intriguing and rather plausible. But I don't know how to make it rigorous, in terms of the standard presentation of intersection theory. (And neither does an algebraic-geometer friend, who points out various issues, for instance, "... how it generalizes to higher dimensions: would the self-intersection of a surface in a threefold be a generic curve on that surface?")
So, is any rigorous version of this statement available? Failing this, does there even exist any discussion or mention of this heuristic, other than in that Wikipedia article?