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. (The same definition applies to a complex manifold X.) Let D be a finite set of closed subvarieties of X (understood to be irreducible), written formally as a sum, D = ∑ j = 1 r D j {\displaystyle D=\sum _{j=1}^{r}D_{j}} . For some purposes, one may identify D with the closed subset ∪ j D j {\displaystyle \cup _{j}D_{j}} of X. Then D is a divisor with simple normal crossings (or an snc divisor) in X if
X is smooth over k, each D j {\displaystyle D_{j}} is smooth and of codimension 1 in X, and the varieties D j {\displaystyle D_{j}} intersect transversely in X. That is, at a point p that lies on s of the varieties D j {\displaystyle D_{j}} , the intersection of the tangent spaces of those D j {\displaystyle D_{j}} 's at p has codimension s in the tangent space of X at p.
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