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Smooth Variety

A smooth variety over a field K is an algebraic variety whose structure morphism to Spec(K) is a smooth morphism. Over a perfect field, this is equivalent to all local rings of the algebraic variety being regular. Smooth varieties are also called nonsingular varieties.

The points at which an algebraic variety is smooth form its smooth locus. The remaining points are its singular points. In characteristic zero, a resolution of singularities replaces a algebraic variety by a smooth variety through a proper morphism that is also a birational morphism.

See also

Algebraic Variety, Birational Morphism, Proper Morphism, Resolution of Singularities, Singular Point, Smooth Morphism, Structure Morphism

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References

Hartshorne, R. Algebraic Geometry. New York: Springer-Verlag, 1977.The Stacks Project Authors. "Schemes Smooth over Fields." §33.25 in The Stacks Project, Tag 04QM, 2026. https://stacks.math.columbia.edu/tag/04QM.

Cite this as:

Weisstein, Eric W. "Smooth Variety." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SmoothVariety.html

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