Exceptional divisor

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

More precisely, suppose that $f:\; X\; ightarrow\; Y$ 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 $sigma:\; ilde\{X\}\; ightarrow\; X$ of a subvariety $W\; subset\; X$: in this case the exceptional divisor is exactly the preimage of $W$.

References

*cite book | author=Shafarevich, Igor | title=Basic Algebraic Geometry, Vol. 1
publisher=Springer-Verlag | year=1994 | id=ISBN 3-540-54812-2