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).
In algebraic geometry, normal crossing divisors are a class of divisors which generalize the smooth divisors. Intuitively they cross only in a transversal way.
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.
January 06, 2023
normal, crossing, singularity, algebraic, geometry, normal, crossing, singularity, singularity, similar, union, coordinate, hyperplanes, term, confusing, because, normal, crossing, singularities, usually, normal, schemes, sense, local, rings, being, integrally. 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 Contents 1 Normal crossing divisors 2 Normal crossing singularity 3 Simple normal crossing singularity 4 Examples 5 ReferencesNormal crossing divisors EditIn 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 i Z i displaystyle Z bigcup i Z i a reduced Cartier divisor with Z i displaystyle 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 displaystyle Z i are smooth and for each component Z k displaystyle Z k Z Z k Z k displaystyle Z Z k Z k is a smooth normal crossing divisor Equivalently one says that a reduced divisor has normal crossings if each point etale locally looks like the intersection of coordinate hyperplanes Normal crossing singularity EditIn 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 EditIn 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 EditThe 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 x y 0 displaystyle 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 Any variety which is the union of smooth varieties which all have smooth intersections is a variety with normal crossing singularities For example let f g C x 0 x 3 displaystyle f g in mathbb C x 0 ldots x 3 be irreducible polynomials defining smooth hypersurfaces such that the ideal f g displaystyle f g defines a smooth curve Then Proj C x 0 x 3 f g displaystyle text Proj mathbb C x 0 ldots x 3 fg is a surface with normal crossing singularities References EditRobert Lazarsfeld Positivity in algebraic geometry Springer Verlag Berlin 1994 Retrieved from https en wikipedia org w index php title Normal crossing singularity amp oldid 766158164, wikipedia, wiki, book, books, library,