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Smooth Locus

The smooth locus of a scheme X over a base scheme S is the set of points at which the structure morphism X->S is a smooth morphism. For a locally finitely presented morphism X->S, the smooth locus is an open set.

For an algebraic variety over a field, the complement of the smooth locus consists of its singular points. A smooth variety is one whose smooth locus is the whole algebraic variety.

See also

Algebraic Variety, Locally Finitely Presented Morphism, Singular Point, Smooth Morphism, Smooth Variety, Structure Morphism

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References

Grothendieck, A. and Dieudonné, J. "Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, quatrième partie." Publ. Math. IHES 32, 5-361, 1967. https://doi.org/10.1007/BF02732123.The Stacks Project Authors. "Smooth Morphisms." §29.35 in The Stacks Project, Tag 01V4, 2026. https://stacks.math.columbia.edu/tag/01V4.

Cite this as:

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

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