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

References

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