In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map
f:X → Y
of varieties is a kind of 'large' subvariety of
X
f
More precisely, suppose that
f:X → Y
is a regular map of varieties which is birational (that is, it is an isomorphism between open subsets of
X
Y
Z\subsetX
f(Z)
Y
f
\sumiZi\inDiv(X),
where the sum is over all exceptional subvarieties of
f
X
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
\sigma:\tilde{X} → X
of a subvariety
W\subsetX
in this case the exceptional divisor is exactly the preimage of
W