In SGA 1 Expose XIII, $\S2.1$, they give a definition of a "normal crossings divisor relative to $S$" which is a bit difficult to parse (and has some typos).

My first question is: Is the following a correct interpretation of their definition? (If not, what is the right definition?)

Let $f : X\rightarrow S$ be a morphism of schemes, and let $D$ be an effective cartier divisor on $X$. We say that it is a strict normal crossings divisor relative to $S$ if for every $x\in\text{Supp }D$,

  1. $X$ is smooth over $S$ at $x$,

  2. there exist $f_1,\ldots,f_r\in \mathcal{O}_{X,x}$ such that the ideal $I_{D,x}$ of $\mathcal{O}_{X,x}$ corresponding to $D$ is generated by $\prod_{i=1}^r f_i$ and such that the ideal $(f_1,f_2,\ldots,f_r)$ has height $r$ (ie, the closed subscheme given by $(f_1,f_2,\ldots,f_r)$ is pure codimension $r$ in $\text{Spec }\mathcal{O}_{X,x}$), and

  3. letting $s := f(x)$, and $Y := \text{Spec }\mathcal{O}_{X,x}/(f_1,\ldots,f_r)$, then $Y$ is flat over $\mathcal{O}_{S,s}$ and $Y_s$ is a smooth $\kappa(s)$-algebra.

A normal crossings divisor relative to $S$ is then just an effective cartier divisor which etale locally on $X$ (or should it be local on $S$?) is strict NCD over $S$.

Later, in Proposition 5.1, they seem to give a definition of a normal crossings divisor which is compatible with the definition given in the stacks project:

If $X$ is a locally noetherian scheme, then a strict normal crossings divisor on $X$ is an effective Cartier divisor $D\subset X$ such that for every $p\in D$, the local ring $\mathcal{O}_{X,p}$ is regular and there exists a regular system of parameters $f_1,\ldots,f_d\in\mathfrak{m}_p$ and $1\le r\le d$ such that $D$ is cut out by $f_1,\ldots,f_r$ in $\mathcal{O}_{X,p}$. A normal crossings divisor is then an effective Cartier divisor which etale locally is a strict NCD.

I'm fairly confident this definition of (non-relative) NCD's is correct.

My second question is: Consider $X = \text{Spec }\mathbb{Z}[t]$, and let $D$ be the divisor $t(t-2)$. Am I correct in saying that this is a strict normal crossings divisor, but is not a strict NCD relative to $\text{Spec }\mathbb{Z}$? (it fails the flatness part of condition (3)). This seems slightly weird to me. For example, $D$ is certainly a relative effective Cartier divisor (relative to $\text{Spec }\mathbb{Z}$) since it is flat over $\mathbb{Z}$, and $D$ is also a strict NCD, but is not a strict NCD relative to $\mathbb{Z}$. Would it be accurate to say that the "relative" part of being a relative NCD is that the "crossings should also be relative?" (ie, flat over $S$)?

  • $\begingroup$ 1) I think the aim of introducing "strictly of normal crossings" is just to have a model for the local behaviour of "normal crossings"; so it is perfectly correct. 2) In the case $|I|=1$, the definition tells us that $D$ has normal crossings over $S$ if and only if it is smooth over $S$; what's the problem? $\endgroup$
    – abx
    Aug 1, 2014 at 12:20
  • 1
    $\begingroup$ The model for "relative ncd" is a union of coordinate hyperplanes in an affine space over $S$. It must etale-local at each $x\in D$ since only for the etale topology do smooth $S$-schemes look like affine spaces. But irreducibility is not etale-local! A nodal plane curve over a field $k$ is a relative ncd over $k$ since etale-locally at the non-smooth point $z$ it becomes two $k$-smooth curves crossing transversally, but that description cannot be attained Zariski-locally. Note that $I(z)$ of size 2 is determined only etale-locally at $z$! The interest lies in non-smooth part. $\endgroup$
    – user27920
    Aug 1, 2014 at 13:18
  • $\begingroup$ So am I correct in saying that a section of a smooth curve $X/S$ determines a NCD relative to S? $\endgroup$
    – Will Chen
    Aug 1, 2014 at 14:19
  • 3
    $\begingroup$ @oxeimon: The whole setup is relative over $S$, so most definitely $S$ can be arbitrarily "bad"; this is the entire purpose of Grothendieck's relative approach to algebraic geometry. Anyway, the business about the affine spaces refers to the local structure theorem for smooth morphisms: if $f:X \rightarrow S$ is smooth then every $x \in X$ admits a Zariski-open neighborhood $U$ for which there is an etale $S$-morphism $U \rightarrow \mathbf{A}^n_S$. This is explained very nicely in section 17 of EGA IV$_4$, and more accessibly in Chapter 2 of the book "Neron models". $\endgroup$
    – user27920
    Aug 1, 2014 at 15:13
  • 3
    $\begingroup$ For your first question: yes, a section of a smooth curve is NCD relative to the base, no matter how bad the base is. For the second, have a look at EGA IV 17.15.3. $\endgroup$
    – abx
    Aug 1, 2014 at 15:13


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.