View source for Normal crossing singularity

In [[Algebraic geometry|algebraic geometry]] a '''normal crossing singularity''' is a singularity similar to a union of coordinate hyperplanes. The term can be confusing because normal crossing singularities are not usually normal schemes (in the sense of the local rings being integrally closed).
==Normal crossing divisors==
In [[Algebraic geometry|algebraic geometry]], '''normal crossing divisors''' are  a class of [[Divisor (algebraic geometry)|divisors]] which generalize the smooth divisors. Intuitively they cross only in a transversal way.

Let ''A'' be an [[Algebraic variety|algebraic variety]], and <math>Z= \bigcup_i Z_i</math> a [[Divisor (algebraic geometry)|reduced Cartier divisor]], with <math>Z_i</math> its irreducible components. Then ''Z'' is called '''a smooth normal crossing divisor''' if either

:(i) ''A'' is a [[Algebraic curve|curve]], or 
:(ii) all <math>Z_i</math> are smooth, and for each component <math>Z_k</math>, <math>(Z-Z_k)|_{Z_k}</math> is a smooth normal crossing divisor.

Equivalently, one says that a reduced divisor has normal crossings if each point [[Étale topology|étale locally]] looks like the intersection of coordinate hyperplanes.

==Normal crossing singularity==
In [[Algebraic geometry|algebraic geometry]] a '''normal crossings ''' singularity is a point in an [[Algebraic variety|algebraic variety]] that is locally isomorphic to a normal crossings divisor.

==Simple normal crossing singularity==
In [[Algebraic geometry|algebraic geometry]] a '''simple normal crossings singularity''' is a point in an [[Algebraic variety|algebraic variety]], the latter having smooth [[Irreducible component|irreducible component]]s, that is locally isomorphic to a normal crossings divisor.

==Examples==
* The normal crossing points in the algebraic variety called the [[Whitney umbrella]] are not simple normal crossings singularities.
* The origin in the algebraic variety defined by <math>xy=0</math> is a simple normal crossings singularity. The variety itself, seen as a subvariety of the two-dimensional [[Cartesian coordinate system|affine plane]] is an example of a normal crossings divisor.
* Any variety which is the union of smooth varieties which all have smooth intersections is a variety with normal crossing singularities. For example, let <math>f,g \in \mathbb{C}[x_0,\ldots,x_3]</math> be irreducible polynomials defining smooth hypersurfaces such that the ideal <math>(f,g)</math> defines a smooth curve. Then <math>\text{Proj}(\mathbb{C}[x_0,\ldots,x_3]/(fg))</math> is a surface with normal crossing singularities.

== References ==

* Robert Lazarsfeld, ''Positivity in algebraic geometry'', Springer-Verlag, Berlin, 1994.

[[Category:Algebraic geometry]]
[[Category:Geometry of divisors]]

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