Cone (algebraic geometry)

In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, the relative Spec C = Spec X ⁡ R {\displaystyle C=\operatorname {Spec} _{X}R} of a quasi-coherent graded OX-algebra R is called the cone or affine cone of R. Similarly, the relative Proj P ( C ) = Proj X ⁡ R {\displaystyle \mathbb {P} (C)=\operatorname {Proj} _{X}R} is called the projective cone of C or R. Note: The cone comes with the G m {\displaystyle \mathbb {G} _{m}} -action due to the grading of R; this action is a part of the data of a cone (whence the terminology).

Source: Wikipedia — Cone (algebraic geometry) (CC BY-SA 4.0)

Cone (algebraic geometry)

In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, the relative Spec C = Spec X ⁡ R {\displaystyle C=\operatorname {Spec} _{X}R} of a quasi-coherent graded OX-algebra R is called the cone or affine cone of R. Similarly, the relative Proj P ( C ) = Proj X ⁡ R {\displaystyle \mathbb {P} (C)=\operatorname {Proj} _{X}R} is called the projective cone of C or R. Note: The cone comes with the G m {\displaystyle \mathbb {G} _{m}} -action due to the grading of R; this action is a part of the data of a cone (whence the terminology).

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Source: Wikipedia "Cone (algebraic geometry)" · CC BY-SA 4.0

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