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 "O""X"-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 formula_3-action due to the grading of "R"; this action is a part of the data of a cone (whence the terminology).
Examples.
Computations.
Consider the complete intersection ideal formula_25 and let formula_26 be the projective scheme defined by the ideal sheaf formula_27. Then, we have the isomorphism of formula_28-algebras is given by
Properties.
If formula_30 is a graded homomorphism of graded "O""X"-algebras, then one gets an induced morphism between the cones:
If the homomorphism is surjective, then one gets closed immersions formula_32
In particular, assuming "R"0 = "O""X", the construction applies to the projection formula_33 (which is an augmentation map) and gives
It is a section; i.e., formula_35 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 formula_37. The projective cone formula_38 is called the projective completion of "C""R". Indeed, the zero-locus "t" = 0 is exactly formula_39 and the complement is the open subscheme "C""R". The locus "t" = 0 is called the hyperplane at infinity.
"O"(1).
Let "R" be a quasi-coherent graded "O""X"-algebra such that "R"0 = "O""X" and "R" is locally generated as "O""X"-algebra by "R"1. 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 "x""i"'s. Thus,
Then formula_42 has the line bundle "O"(1) given by the hyperplane bundle formula_43 of formula_44; gluing such local "O"(1)'s, which agree locally, gives the line bundle "O"(1) on formula_45.
For any integer "n", one also writes "O"("n") for the "n"-th tensor power of "O"(1). If the cone "C"=Spec"X""R" 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.