Normal crossing singularity

In geometry or more specifically, in algebraic geometry normal crossings refers to 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

References

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