Exceptional divisor

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


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