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).
Consider the complete intersection ideal
be the projective scheme defined by the ideal sheaf
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: explicitly, the n-th degree piece is Then the affine cone of it is denoted by
is called the projective completion of CR.
and the complement is the open subscheme CR.
The locus t = 0 is called the hyperplane at infinity.
Let R be a quasi-coherent 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.
; 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.