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I could not find any reference for normal crossings. The definition here is not so clear to me. In some texts, they sometimes said that two varieties have normal-crossing (non-normal crossing) with singularity .... Could some one tell me what exactly this means? For examples, what does it mean if two varieties $V(f)$ and $V(g)$ where $f,g$ are two polynomials, have normal crossings?

Many thanks in advance.

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    $\begingroup$ Take a look at Mustata's notes on page 69: math.lsa.umich.edu/~mmustata/lecture_notes_birational.pdf $\endgroup$
    – rfauffar
    Commented Feb 17, 2014 at 21:04
  • $\begingroup$ Thank Robert Auffarth. Could you give me some easy example with illustration? $\endgroup$
    – 9999
    Commented Feb 17, 2014 at 21:18
  • $\begingroup$ Dear @RobertAuffarth, that seems like a good answer. Why not write it in the answer box? $\endgroup$
    – user64687
    Commented Feb 18, 2014 at 9:14

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Mustata's notes on page 69 give a good definition: math.lsa.umich.edu/~mmustata/lecture_notes_birational.pdf.

Basically if your variety is $n$-dimensional, you want a divisor whose irreducible components are smooth and that intersect each other at any given point like (at most $n$) hyperplanes would intersect each other. For example, in dimension 3, the following picture could locally represent a SNC divisor, where the lines actually represent smooth irreducible divisors:

enter image description here

Edit: This is the definition of a simple normal crossing divisor.

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  • $\begingroup$ Dear Robert, just for clarity let me ask: are you describing simple normal crossings divisors? $\endgroup$
    – user64687
    Commented Feb 18, 2014 at 13:28
  • $\begingroup$ Yes you're right, it seems the difference (according to wikipedia) lies in the condition that at most $n$ components meet at a point. I'm not positive, however, since the wikipedia page given above isn't well written. $\endgroup$
    – rfauffar
    Commented Feb 18, 2014 at 14:38
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    $\begingroup$ Dear Robert, yes, the Wikipedia page is not clear. In fact the difference is that simple normal crossings divisors must have smooth components, while normal crossings divisors need not. So for instance a nodal plane curve is a NC divisor, but not an SNC divisor. A reference is Kollár, Lectures on Resolution of Singularities, pg. 30. $\endgroup$
    – user64687
    Commented Feb 18, 2014 at 15:29
  • $\begingroup$ Dear @AsalBeagDubh, thank you very much for your clarification, it makes a lot of sense. $\endgroup$
    – rfauffar
    Commented Feb 18, 2014 at 15:30
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    $\begingroup$ Mircea has taken this down, it looks like :( $\endgroup$
    – Hoot
    Commented Aug 23, 2015 at 4:04

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