Cone (algebraic geometry)

In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, the relative Spec

${\displaystyle C=\operatorname {Spec} _{X}R}$

of a quasi-coherent graded O_X-algebra R is called the cone or affine cone of R. Similarly, the relative Proj

${\displaystyle \mathbb {P} (C)=\operatorname {Proj} _{X}R}$

is called the projective cone of C or R.

Note: The cone comes with the ${\displaystyle \mathbb {G} _{m}}$-action due to the grading of R; this action is a part of the data of a cone (whence the terminology).