Cone (algebraic geometry)

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

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

is called the projective cone of C or R.

Note: The cone comes with the -action due to the grading of R; this action is a part of the data of a cone (whence the terminology).



If is a graded homomorphism of graded OX-algebras, then one gets an induced morphism between the cones:


If the homomorphism is surjective, then one gets closed immersions

In particular, assuming R0 = OX, the construction applies to the projection (which is an augmentation map) and gives


It is a section; i.e., is the identity and is called the zero-section embedding.

Consider the graded algebra R[t] with variable t having degree one. Then the affine cone of it is denoted by . The projective cone is called the projective completion of CR. Indeed, the zero-locus t = 0 is exactly and the complement is the open subscheme CR. The locus t = 0 is called the hyperplane at infinity.


Let R be a graded OX-algebra such that R0 = OX and R is locally generated as OX-algebra by R1. Then, by definition, the projective cone of R is:

where the colimit runs over open affine subsets U of X. By assumption R(U) has finitely many degree-one generators xi's. Thus,

Then has the line bundle O(1) given by the hyperplane bundle of ; gluing such local O(1)'s, which agree locally, gives the line bundle O(1) on .

For any integer n, one also writes O(n) for the n-th tensor power of O(1). If the cone C=SpecXR is the total space of a vector bundle E, then O(-1) is the tautological line bundle on the projective bundle P(E).

Remark: When the (local) generators of R have degree other than one, the construction of O(1) still goes through but with a weighted projective space in place of a projective space; so the resulting O(1) is not necessarily a line bundle. In the language of divisor, this O(1) corresponds to a Q-Cartier divisor.


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