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).
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Transcription
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 a reduced Cartier divisor, with its irreducible components. Then Z is called a smooth normal crossing divisor if either
- (i) A is a curve, or
- (ii) all are smooth, and for each component , 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 crossing 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 crossing 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 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 be irreducible polynomials defining smooth hypersurfaces such that the ideal defines a smooth curve. Then is a surface with normal crossing singularities.
References
- Robert Lazarsfeld, Positivity in algebraic geometry, Springer-Verlag, Berlin, 1994.
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