In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map
of varieties is a kind of 'large' subvariety of
More precisely, suppose that
is a regular map of varieties which is birational (that is, it is an isomorphism between open subsets of
where the sum is over all exceptional subvarieties of
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
of a subvariety
in this case the exceptional divisor is exactly the preimage of