algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

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 algebraic geometry, an algebraic variety or scheme ''X'' is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety ''X'' (understood to be irreducible) is normal if and o ...

(in the sense of the local rings being integrally closed).

Normal crossing divisors

, 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. Mo ...

, 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 In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

, or :(ii) all

Z_i

are smooth, and for each component

Z_k

(Z-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 crossing singularity

a normal crossings singularity is a point in an

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). Pr ...

isomorphic to a normal crossings divisor.

Simple normal crossing singularity

a simple normal crossings singularity is a point in an

, the latter having smooth irreducible components, that is

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

xy=0

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 reals equipped with a metric, the Euclidean distance. In other words, an affine pl ...

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 \in \mathbb_0,\ldots,x_3 /math> be irreducible polynomials defining smooth hypersurfaces such that the ideal (f,g) defines a smooth curve. Then \text(\mathbb{C}_0,\ldots,x_3 (fg)) is a surface with normal crossing singularities.

References

* Robert Lazarsfeld, ''Positivity in algebraic geometry'', Springer-Verlag, Berlin, 1994. Algebraic geometry Geometry of divisors