A smooth variety over a field 
is an algebraic variety
whose structure morphism to 
is a smooth morphism.
Over a perfect field, this is equivalent to all
local rings of the variety being regular. Smooth varieties
are also called nonsingular varieties.
The points at which a variety is smooth form its smooth locus. The remaining points are its singular points.
In characteristic zero, a resolution of
singularities replaces a 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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