# Normal crossing singularity

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Normal_crossing_singularity
> Markdown URL: https://mediated.wiki/source/Normal_crossing_singularity.md
> Source: https://en.wikipedia.org/wiki/Normal_crossing_singularity
> Source revision: 1339173572
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Singularities of algebraic varieties}}
In [algebraic geometry](/source/algebraic_geometry), a '''normal crossing singularity''' looks locally like a union of coordinate [hyperplane](/source/hyperplane)s. There are two variants of the concept, a [divisor](/source/Divisor_(algebraic_geometry)) with '''normal crossings''' or with '''simple normal crossings'''. These can be considered the simplest kind of singularities. Several theorems on [resolution of singularities](/source/resolution_of_singularities) relate an arbitrary variety to a divisor with simple normal crossings in a [smooth variety](/source/smooth_variety).

==Divisor with simple normal crossings==
Let ''X'' be an [algebraic variety](/source/algebraic_variety) over a [perfect field](/source/perfect_field) ''k''. (The same definition applies to a [complex manifold](/source/complex_manifold) ''X''.) Let ''D'' be a finite set of closed subvarieties of ''X'' (understood to be [irreducible](/source/irreducible)), written formally as a sum, <math>D=\sum_{j=1}^r D_j</math>. For some purposes, one may identify ''D'' with the closed subset <math>\cup_j D_j</math> of ''X''. Then ''D'' is a '''divisor with simple normal crossings''' (or an '''snc divisor''') in ''X'' if
*''X'' is smooth over ''k'',
*each <math>D_j</math> is smooth and of [codimension](/source/codimension) 1 in ''X'', and
*the varieties <math>D_j</math> intersect [transversely](/source/transversality) in ''X''. That is, at a point ''p'' that lies on ''s'' of the varieties <math>D_j</math>, the intersection of the [tangent space](/source/tangent_space)s of those <math>D_j</math>'s at ''p'' has codimension ''s'' in the tangent space of ''X'' at ''p''.
thumb|right|200px|A divisor with simple normal crossings in the affine plane

The transversality condition can be rephrased in several ways. Over the complex numbers, it is equivalent to say that at a complex point ''p'' that lies on ''s'' of the subvarieties, say <math>D_1,\ldots,D_s</math>, there is a [complex analytic](/source/complex_analytic) [coordinate chart](/source/Atlas_(topology)) around ''p'' in which ''p'' is the origin in <math>\mathbb{C}^n=\{(z_1,\ldots,z_n): z_j\in \mathbb{C}\text{ for each }j \}</math> and <math>D_j</math> is the coordinate hyperplane <math>\{ z_j=0 \}</math>, for <math>j=1,\ldots,s</math>.<ref>Lazarsfeld (2004), Definition 4.1.1.</ref> In the language of [schemes](/source/scheme_(mathematics)), transversality means that the [scheme-theoretic intersection](/source/scheme-theoretic_intersection) of any ''s'' of the <math>D_j</math>'s is smooth of codimension ''s'' in ''X'' (or empty).<ref>{{Citation | title=Stacks Project, Tag 0BIA | url=http://stacks.math.columbia.edu/tag/0BIA}}.</ref>

Outside the setting of varieties over a perfect field, the following more general definition is used. Let ''X'' be a scheme, <math>D=\sum_j D_j</math> a formal sum of [integral](/source/integral_scheme) closed subschemes. For each point ''p'' in ''X'', let <math>O_{X,p}</math> be the [local ring](/source/local_ring) of ''X'' at ''p'' (the ring of regular functions near ''p''), with maximal ideal <math>\mathfrak{m}</math> (the functions vanishing at ''p'') and [residue field](/source/residue_field) <math>k(p)</math>. Say that functions <math>z_1,\ldots,z_n</math> in <math>\mathfrak{m}</math> form ''local coordinates'' at ''p'' if they map to a [basis](/source/basis_(linear_algebra)) for the <math>k(p)</math>-vector space <math>\mathfrak{m}/\mathfrak{m}^2</math>. Then ''D'' is a '''divisor with simple normal crossings''' in ''X'' if ''X'' is [regular](/source/regular_scheme) and for each point ''p'' in ''X'', there are local coordinates <math>z_1,\ldots,z_n</math> at ''p'' for which each <math>D_j</math> that contains ''p'' is equal to the closed subscheme <math>\{ z_{i(j)}=0 \}</math> near ''p'' for some <math>i(j)</math>.<ref name="kollar17">Kollár (2013), Definition 1.7.</ref>

There is a more general notion of a divisor, meaning a formal sum of codimension-1 subvarieties with integer coefficients, <math>D=\sum_{j=1}^r a_j D_j</math>. A divisor ''D'' is said to have simple normal crossings in ''X'' if the associated "reduced" divisor <math>\sum_{j=1}^r D_j</math> has simple normal crossings in ''X''.<ref name="kollar17" />

==Resolution of singularities==
Although a divisor with simple normal crossings is very special, the concept can be used to study arbitrary varieties using [Heisuke Hironaka](/source/Heisuke_Hironaka)'s theorems on '''resolution of singularities'''. One result is: let ''X'' be a variety over a field of [characteristic](/source/characteristic_(algebra)) zero, and let ''S'' be a [Zariski closed](/source/Zariski_closed) subset that contains the singular locus of ''X'' and is not all of ''X''. (The case where ''S'' is equal to the singular locus is already important.) Then there is a [proper](/source/proper_morphism) [birational morphism](/source/birational_morphism) ''f'' from a smooth variety ''Y'' to ''X'' such that ''f'' is an isomorphism over ''X'' – ''S'' and the inverse image of ''S'' is a divisor with simple normal crossings in ''Y''.<ref>Kollár (2007), Theorems 3.26 and 3.27.</ref> This is an optimal statement; one cannot always make the inverse image of ''S'' smooth, for example.

[Alexander Grothendieck](/source/Alexander_Grothendieck) conjectured that the same thing (in terms of regular schemes rather than smooth varieties) should be true for algebraic varieties over any field, and even more generally, for [quasi-excellent scheme](/source/quasi-excellent_scheme)s.

==Divisor with normal crossings==
More generally, <math>D=\sum_j D_j</math> is a '''divisor with normal crossings''' in a scheme ''X'' if ''X'' is regular and for every point ''p'' in ''X'', there is an [étale morphism](/source/%C3%A9tale_morphism) <math>X'\to X</math> with ''p'' in the image such that the inverse image of ''D'' is a divisor with simple normal crossings in <math>X'</math>.<ref name="kollar17" /> When ''X'' is a variety over a perfect field ''k'', it is equivalent to say that the inclusion of ''D'' into ''X'' is [étale-locally](/source/%C3%A9tale_topology) isomorphic to a union of coordinate hyperplanes in affine space <math>A^n_k</math>. A divisor with normal crossings has simple normal crossings if and only if each irreducible component of ''D'' is regular.

==Examples==
* The closed subset <math>\{ xy=0 \}</math> in the affine plane <math>A^2</math> over a field, viewed as a divisor, has simple normal crossings. This is the union of the two coordinate axes. 100px
* The nodal cubic curve <math>D=\{ y^2=x^2(x+1) \}</math> is a divisor with normal crossings in the affine plane, but it does not have simple normal crossings. (Simple normal crossings would imply that each irreducible component of ''D'' is regular, whereas in this case ''D'' is irreducible and singular.) 75px
* The cuspidal cubic curve <math>D=\{ y^2=x^3 \}</math> in the affine plane does not have normal crossings. 75px
* The divisor <math>\{ (y+x^2)(y-x^2)=0 \}</math> in the affine plane, say over <math>\mathbb{C}</math>, does not have normal crossings. (The two irreducible components are smooth, but they do not intersect transversely.) 75px
* The divisor <math>\{ yx(y-x)=0 \}</math> in the affine plane does not have normal crossings. (Each of the three irreducible components is smooth, and any two of them intersect transversely, but the three together are not transverse. Transversality would imply that the intersection of more than ''n'' components in an ''n''-dimensional variety is empty.) 75px

==Normal crossing scheme==
A scheme ''Y'' is said to be '''snc''' or to have '''simple normal crossings''' if every point has a Zariski open neighborhood which is isomorphic to a divisor with simple normal crossings (viewed as a [reduced](/source/reduced_scheme) closed subscheme) in some regular scheme. Thus ''Y'' need not be given (globally) as a closed subscheme of a regular scheme.<ref>Kollár (2013), Definition 1.8.</ref> Likewise, a scheme ''Y'' has '''normal crossings''' if every point has an étale neighborhood <math>Y'\to Y</math> such that <math>Y'</math> is isomorphic to a divisor with simple normal crossings in some regular scheme. For example, [stable curve](/source/stable_curve)s are normal crossing schemes of dimension 1. 

==Notes==
{{reflist|30}}

==References==
*{{Citation | author1-first=János | author1-last=Kollár | author1-link=János Kollár | title=Lectures on Resolution of Singularities | publisher=[Princeton University Press](/source/Princeton_University_Press) | year=2007 | isbn=978-0-691-12923-5 | mr=2289519}}
*{{Citation | author1-first=János | author1-last=Kollár | author1-link=János Kollár | title=Singularities of the Minimal Model Program | publisher=[Cambridge University Press](/source/Cambridge_University_Press) | year=2013 | isbn=978-1-107-03534-8 | mr=3057950 |doi=10.1017/CBO9781139547895}}
*{{Citation| title = Positivity in Algebraic Geometry (2 vols.) | last = Lazarsfeld | first = Robert | year = 2004 | author-link = Robert Lazarsfeld | publisher = Springer-Verlag | location = Berlin | doi = 10.1007/978-3-642-18808-4 | isbn = 3-540-22533-1 | mr = 2095471}}

==External links==
*{{Citation | author1=The Stacks Project Authors | title=The Stacks Project  | url=http://stacks.math.columbia.edu/}}

Category:Singularity theory
Category:Geometry of divisors

---
Adapted from the Wikipedia article [Normal crossing singularity](https://en.wikipedia.org/wiki/Normal_crossing_singularity) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Normal_crossing_singularity?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.