Exceptional divisor

In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map
of varieties is a kind of 'large' subvariety of formula_2 which is 'crushed' by formula_3, 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 formula_2 and formula_6). A codimension-1 subvariety formula_7 is said to be "exceptional" if formula_8 has codimension at least 2 as a subvariety of formula_6. One may then define the "exceptional divisor" of formula_3 to be
where the sum is over all exceptional subvarieties of formula_3, and is an element of the group of Weil divisors on formula_2.
Consideration of exceptional divisors is crucial in birational geometry: an elementary result (see for instance Shafarevich, II.4.4) shows (under suitable assumptions) that any birational regular map that is not an isomorphism has an exceptional divisor. A particularly important example is the blowup
of a subvariety
in this case the exceptional divisor is exactly the preimage of formula_16.