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
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
which generalize the smooth divisors. Intuitively they cross only in a transversal way.
Let ''A'' be an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
, 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
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
, 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
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
that is
locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points).
P ...
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
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
, the latter having
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
irreducible component
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal ( ...
s, that is
locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points).
P ...
isomorphic to a normal crossings divisor.
Examples
* The normal crossing points in the algebraic variety called the
Whitney umbrella
frame, Section of the surface
In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface placed in three dime ...
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
In geometry, an affine plane is a two-dimensional affine space.
Examples
Typical examples of affine planes are
*Euclidean planes, which are affine planes over the real number, reals equipped with a metric (mathematics), metric, the Euclidean dista ...
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