Divisors

Divisors encode codimension-one geometric data with multiplicities under specified definitions.

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Algebraic singularities
Singular points fail suitable smoothness conditions measured by local geometry or tangent behavior.
Sheaves
A sheaf organizes local data that restrict consistently and glue uniquely when compatible.
Affine varieties
Common polynomial zero sets define affine algebraic sets over a chosen field.
Algebraic dimension
Chains of prime ideals define Krull dimension and connect algebraic structure to geometric dimension.
Coordinate rings
Polynomial functions on an affine algebraic set form a quotient by its vanishing ideal.