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In 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

• Normal crossing singularity

• Simple normal crossing singularity

• Examples

• References

Normal crossing divisors are a class of divisors which generalize the smooth divisors. Intuitively they cross only in a transversal way.

Let A be an algebraic variety, and Z = ⋃ ⋃ i Z i {\displaystyle Z=\bigcup _{i}Z_{i}} a reduced Cartier divisor, with Z i {\displaystyle Z_{i}} its irreducible components. Then Z is called a smooth normal crossing divisor if either

(i) A is a curve, or

(ii) all Z i {\displaystyle Z_{i}} are smooth, and for each component Z k {\displaystyle Z_{k}} , ( Z − − Z k ) | Z k {\displaystyle (Z-Z_{k})|_{Z_{k}}} is a smooth normal crossing divisor.

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

A normal crossings singularity is a point in an algebraic variety that is locally isomorphic to a normal crossings divisor.

A simple normal crossings singularity is a point in an algebraic variety, the latter having smooth irreducible components, that is locally isomorphic to a normal crossings divisor.

• 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 x y = 0 {\displaystyle xy=0} is a simple normal crossings singularity. The variety itself, seen as a subvariety of the two-dimensional 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 f , g ∈ ∈ C [ x 0 , … … , x 3 ] {\displaystyle f,g\in \mathbb {C} [x_{0},\ldots ,x_{3}]} be irreducible polynomials defining smooth hypersurfaces such that the ideal ( f , g ) {\displaystyle (f,g)} defines a smooth curve. Then Proj ( C [ x 0 , … … , x 3 ] / ( f g ) ) {\displaystyle {\text{Proj}}(\mathbb {C} [x_{0},\ldots ,x_{3}]/(fg))} is a surface with normal crossing singularities.

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