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Algebraic Blow-Up

An algebraic blow-up of a scheme X along a closed subscheme Z is the scheme equipped with the morphism

b:Bl_(Z)(X)=Proj_(X)( direct sum _(n>=0)I^n)->X,

where I is the ideal sheaf defining Z (Stacks Project 2026). The subscheme Z is called the blow-up center.

The morphism b is an isomorphism away from Z, while the inverse image b^(-1)(Z) is the exceptional divisor. For example, blowing up the affine plane at the origin replaces that point by a projective line recording the directions through the origin.

Algebraic blow-ups are used in the study and resolution of singularities. They are distinct from finite-time blow-up in analysis and from graph blow-ups in graph theory.

See also

Algebraic Variety, Blow-Up, Blow-Up Center, Exceptional Divisor, Ideal Sheaf, Projective Space, Scheme, Subscheme

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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. "Algebraic Blow-Up." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlgebraicBlow-Up.html

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