Concept# Normal crossing singularity

Summary

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
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= \bigcup_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

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