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


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 $S$"?

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    $\begingroup$ You should be aware that the terminology varies a bit. Some people say "normal crossings" where others would say "strict" or "simple" "normal crossings". In the end it usually doesn't matter since you always blow it up to get the stronger condition. $\endgroup$ Jun 29 '11 at 18:40
  • $\begingroup$ This is true, but there are some situations when there is an important difference between nc and snc, for instance in considering the singularities of pairs $(X,D)$. For more details see my answer below. $\endgroup$ Jan 22 '13 at 5:34

Your definition of normal crossings divisor is, as you say, often called a strict normal crossings divisor. People who use this terminology allow normal crossings divisors to have components which are not necessarily smooth; they are only required to looks like a smooth components meeting transversally locally in the analytic topology (if one works over $\mathbb{C}$) or etale locally (in general).

As an example, an irreducible curve in a smooth surface having (only) an ordinary node as a singularity would be called a normal crossing divisor but would not be a strict normal crossings divisor.


Here is the definition of an sncd (and ncd) from SGA (SGA 5, 3.1.5, pg 24):

Let $X$ be a regular scheme and $D$ an effective divisor on $X$.

DEF: $D$ is an sncd on $X$ if there is a finite family of sections $(f_i)_{i\in I}$, $f_i\in O_X(X)$ such that the following two conditions hold:

i) $D = \sum_{i\in I}$ $div(f_i)$

ii) for each $x\in Supp(D)$, the local restrictions $(f_i)_x$ that satisfy $(f_i)_x \in m_x$ (i.e. those that land in the maximal ideal of $O_{X,x}$) have the property that they form a part of a regular system of parameters for the local ring $O_{X,x}$.

DEF: an effective divisor $D$ is an ncd if (i) and (ii) hold 'etale locally.

This definition is unambiguous (unlike Hartshorne, Wikipedia, etc) since, e.g., regularity is not a relative condition (whereas smoothness is a relative condition). Further, you can check that two smooth hyperplanes meeting transversally in an affine neighborhood, e.g. the x-y axis in $A^3_k$ for $k$ a field, form an sncd.

Example / Remark : in SGA 5 they write global sections for an sncd. So here is an example of an ncd that is not an sncd: let $X = \mathbb{P}^1_k$ be the projective line over a field $k$ and consider an affine open $U = \mathbb{A}^1_k = Spec \; k[t]$. Now, the divisor defined by $t$ is not an sncd on $X$ as $t\not\in O_X(X) = k$ but since $U\hookrightarrow X$ is etale, then this divisor is an ncd on $X$.

Perhaps it's a matter of taste whether one defines sncd globally as in SGA 5 or Zariski locally...

(N.B. in EGA IV you can find the definitions of div and Supp)

  • $\begingroup$ With this definition, is it still true that each quasi-projective smooth variety has a smooth compactification by a sncd? The example with $\mathbb{P}^1$ makes me question that. $\endgroup$
    – Arkadij
    Sep 5 '20 at 15:35

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.


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