This question is going to be extremely vague.


It seems that wherever I go (especially about Grothendieck's circle of ideas) the higher-dimensional analogue of a curve minus a finite number of points is a scheme minus a normal crossing divisor.


Why is that? What's so special about a normal crossing divisor that it simulates a curve minus a finite number of points better?



Why are normal crossing divisors nice?
It mostly has to do with finding nice compactifications. Compactifications of
varieties are a good thing as they allow us to control what happens at
"infinity". If the variety itself is smooth it seems a good idea (and it is!) to
demand that the compactification also be smooth. However, you need the situation
to be nice at infinity in order to make the study of asymptotic behaviour at
infinity to be as easy as possible. The best behaviour at infinity would be if
the complement were smooth but that is in general not possible. What is always possible
is to demand that the complement be a divisor with normal crossings. In practice
it works essentially as well as having a smooth complement: You have a bunch of
smooth varieties intersecting in as nice a manner as possible.


