Exceptional locus

In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map

of varieties is a kind of 'large' subvariety of ${\displaystyle X}$ which is 'crushed' by ${\displaystyle f}$, in a certain definite sense. More strictly, f has an associated exceptional locus which describes how it identifies nearby points in codimension one, and the exceptional divisor is an appropriate algebraic construction whose support is the exceptional locus. The same ideas can be found in the theory of holomorphic mappings of complex manifolds.

More precisely, suppose that

is a regular map of varieties which is birational (that is, it is an isomorphism between open subsets of ${\displaystyle X}$ and ${\displaystyle Y}$). A codimension-1 subvariety ${\displaystyle Z\subset X}$ is said to be exceptional if ${\displaystyle f(Z)}$ has codimension at least 2 as a subvariety of ${\displaystyle Y}$. One may then define the exceptional divisor of ${\displaystyle f}$ to be

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