Exceptional divisor

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In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map $f: X \rightarrow Y$ of varieties is a kind of 'large' subvariety of $X$ which is 'crushed' by $f$ , in a certain definite sense.

More precisely, suppose that $f: X \rightarrow Y$ is a regular map of varieties which is birational (that is, it is an isomorphism between open subsets of $X$ and $Y$ ). A codimension-1 subvariety $Z \subset X$ is said to be exceptional if $f (Z)$ has codimension at least 2 as a subvariety of $Y$ . One may then define the exceptional divisor of $f$ to be $\sum_i Z_i \in Div(X)$ , where the sum is over all exceptional subvarieties of $f$ , and is an element of the group of Weil divisors on $X$ .

Consideration of exceptional divisors is crucial in birational geometry: an elementary result (see for instance Shafarevich, II.4.4) shows that any birational regular map that is not an isomorphism has an exceptional divisor. A particularly important example is the blowup $\sigma: \tilde{X} \rightarrow X$ of a subvariety $W \subset X$ : in this case the exceptional divisor is exactly the preimage of $W$ .