Normal crossing singularity


In algebraic geometry, a normal crossing singularity looks locally like a union of coordinate hyperplanes. There are two variants of the concept, a divisor with normal crossings or with simple normal crossings. These can be considered the simplest kind of singularities. Several theorems on resolution of singularities relate an arbitrary variety to a divisor with simple normal crossings in a smooth variety.

Divisor with simple normal crossings

Let X be an algebraic variety over a perfect field k. Let D be a finite set of closed subvarieties of X, written formally as a sum,. For some purposes, one may identify D with the closed subset of X. Then D is a divisor with simple normal crossings in X if
  • X is smooth over k,
  • each is smooth and of codimension 1 in X, and
  • the varieties intersect transversely in X. That is, at a point p that lies on s of the varieties, the intersection of the tangent spaces of those 's at p has codimension s in the tangent space of X at p.
The transversality condition can be rephrased in several ways. Over the complex numbers, it is equivalent to say that at a complex point p that lies on s of the subvarieties, say, there is a complex analytic coordinate chart around p in which p is the origin in and is the coordinate hyperplane, for. In the language of schemes, transversality means that the scheme-theoretic intersection of any s of the 's is smooth of codimension s in X.
Outside the setting of varieties over a perfect field, the following more general definition is used. Let X be a scheme, a formal sum of integral closed subschemes. For each point p in X, let be the local ring of X at p, with maximal ideal and residue field. Say that functions in form local coordinates at p if they map to a basis for the -vector space. Then D is a divisor with simple normal crossings in X if X is regular and for each point p in X, there are local coordinates at p for which each that contains p is equal to the closed subscheme near p for some.
There is a more general notion of a divisor, meaning a formal sum of codimension-1 subvarieties with integer coefficients,. A divisor D is said to have simple normal crossings in X if the associated "reduced" divisor has simple normal crossings in X.

Resolution of singularities

Although a divisor with simple normal crossings is very special, the concept can be used to study arbitrary varieties using Heisuke Hironaka's theorems on resolution of singularities. One result is: let X be a variety over a field of characteristic zero, and let S be a Zariski closed subset that contains the singular locus of X and is not all of X. Then there is a proper birational morphism f from a smooth variety Y to X such that f is an isomorphism over X – S and the inverse image of S is a divisor with simple normal crossings in Y. This is an optimal statement; one cannot always make the inverse image of S smooth, for example.
Alexander Grothendieck conjectured that the same thing should be true for algebraic varieties over any field, and even more generally, for quasi-excellent schemes.

Divisor with normal crossings

More generally, is a divisor with normal crossings in a scheme X if X is regular and for every point p in X, there is an étale morphism with p in the image such that the inverse image of D is a divisor with simple normal crossings in. When X is a variety over a perfect field k, it is equivalent to say that the inclusion of D into X is étale-locally isomorphic to a union of coordinate hyperplanes in affine space. A divisor with normal crossings has simple normal crossings if and only if each irreducible component of D is regular.

Examples

  • The closed subset in the affine plane over a field, viewed as a divisor, has simple normal crossings. This is the union of the two coordinate axes.
  • The nodal cubic curve is a divisor with normal crossings in the affine plane, but it does not have simple normal crossings.
  • The cuspidal cubic curve in the affine plane does not have normal crossings.
  • The divisor in the affine plane, say over, does not have normal crossings.
  • The divisor in the affine plane does not have normal crossings.

    Normal crossing scheme

A scheme Y is said to be snc or to have simple normal crossings if every point has a Zariski open neighborhood which is isomorphic to a divisor with simple normal crossings in some regular scheme. Thus Y need not be given as a closed subscheme of a regular scheme. Likewise, a scheme Y has normal crossings if every point has an étale neighborhood such that is isomorphic to a divisor with simple normal crossings in some regular scheme. For example, stable curves are normal crossing schemes of dimension 1.