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Exceptional Divisor

The exceptional divisor of an algebraic blow-up b:Bl_(Z)(X)->X is the inverse image E=b^(-1)(Z) of the blow-up center Z (Stacks Project 2026). It is an effective Cartier divisor, and the blow-up morphism is an isomorphism away from E.

For example, the exceptional divisor of the algebraic blow-up of the affine plane at the origin is a projective line parametrizing directions through the origin.

See also

Algebraic Blow-Up, Blow-Up Center, Cartier Divisor, Effective Cartier Divisor, Projective Line

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References

Hartshorne, R. Algebraic Geometry. New York: Springer-Verlag, 1977.The Stacks Project Authors. "Blowing Up." §31.33 in The Stacks Project, Tag 01OF, 2026. https://stacks.math.columbia.edu/tag/01OF.

Cite this as:

Weisstein, Eric W. "Exceptional Divisor." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ExceptionalDivisor.html

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