Normal crossings

In algebraic geometry normal crossings is the property of intersecting geometric objects to do it in a transversal way.

Normal crossing divisors

In algebraic geometry, 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 $Z= \cup_i Z_i$ a reduced Cartier divisor, with 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 are smooth, and for each component Z_k, $(Z-Z_k)|_{Z_k}$ 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.

Normal crossings singularity

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

Simple normal crossings singularity

In algebraic geometry 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.

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 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.

References

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