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 and let be the projective scheme defined by the ideal sheaf . Then, we have the isomorphism of -algebras is given by[ citation needed]
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 Rt] with variable t having degree one: explicitly, the n-th degree piece is
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 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. 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.
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 and let be the projective scheme defined by the ideal sheaf . Then, we have the isomorphism of -algebras is given by[ citation needed]
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 Rt] with variable t having degree one: explicitly, the n-th degree piece is
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 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. 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.