TITLE: normal crossing divisor v.s. strict normal crossing divisor
QUESTION [9 upvotes]: On wikipedia, the normal crossing divisor is defined to be (by my understanding):
(Assume $X/k$ be a smooth geometrically integral scheme of finite type over a field $k$).
Let $D = \sum_{i=1}^n C_i$ be a Weil Divisor, here $C_i$ are irreducible closed subsets of codimension 1 of $X$. Endow $C_i$ with the reduced scheme structure (hence they are integral closed $k$-scheme of $X$ of codimension 1.) We call $D$ is a normal crossing divisor if each $D_i$ is smooth over $k$ and $D_i$'s intersect transversely.
But in somewhere, I saw the notion "strict normal crossing divisor". For example,
Definition 1.5.1, p.8, in http://math.arizona.edu/~swc/aws/07/KedlayaNotes10Mar.pdf
or
5.1, p.16, in http://www.uni-due.de/~bm0032/publ/TubNbdDocumenta.pdf
The definition looks the same as "normal crossing divisor".
I would like to know what's the difference between these two definitions. It 
would be great with examples. Also, if the base scheme $S$ is not a field, then is the definition the same as above with the modification "each $D_i$ is smooth over $S$"?

REPLY [3 votes]: The definition you gave is currently the accepted definition of snc. As a friend of mine is fond to say: "...and of course, everything you read on wikipedia is correct..."
Anyway, 
this definition essentially means that $D$ is supposed to look like the intersection of coordinate hyperplanes locally at any point. The key is what locally means. In the definition of snc it means Zariski locally, but if you replace that by étale locally, then you get the notion of nc.
As Donu remarked the difference between nc and snc is often negligable. For instance if you want to talk about a resolution of singularities where the preimage of the singular set is nc, then you can usually assume snc, because it just means replacing your resolution with a further blow-up. So, probably people working with those kind of things tend not to worry about the difference. 
On the other hand there is at least one place where the difference is important. 
This is when you want to understand the singularities of a pair $(X,D)$. For simplicity let's say that $X$ is smooth and $D$ is a reduced integral divisor on $X$. Then $(X,D)$ has dlt singularities if and only if $D$ is snc. In particular, an nc pair has "worse" singularities than an "snc" pair. There's a lot more one could say about this, but since this kind of goes out of the scope of the original question I'll stop here.
Remark I realize that you might not know what dlt singularities mean, but in some sense it is not important at the moment. It is an important class of singularities. You can read about them in Kollár-Mori: Birational Geometry of Algebraic Varieties and/or Kollár
Singularities of the Minimal Model Program.