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 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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