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.
More generally, given a vector bundle E on X, if R=Sym is the symmetric algebra generated by the dual of E, then the cone is the total space of E, often written just as E, and the projective cone is the projective bundle of E, which is written as.
Let R be a graded -algebra such that and is coherent and locally generates R as -algebra. Then there is a closed immersion
Computations
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
Properties
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 and gives It is a section; i.e., is the identity and is called the zero-section embedding. Consider the graded algebra R 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 subschemeCR. The locus t = 0 is called the hyperplane at infinity.
''O''(1)
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 has finitely many degree-one generators xi's. Thus, Then has the line bundle O given by the hyperplane bundle of ; gluing such local Os, which agree locally, gives the line bundle O on. For any integer n, one also writes O for the n-th tensor power of O. If the cone C=SpecXR is the total space of a vector bundle E, then O is the tautological line bundle on the projective bundle P. Remark: When the generators of R have degree other than one, the construction of O still goes through but with a weighted projective space in place of a projective space; so the resulting O is not necessarily a line bundle. In the language of divisor, this O corresponds to a Q'-Cartier divisor.
