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

Examples

• If X = Spec k is a point and R is a homogeneous coordinate ring, then the affine cone of R is the (usual) affine cone over the projective variety corresponding to R.
• If for some ideal sheaf I, then is the normal cone to the closed scheme determined by I.
• If for some line bundle L, then is the total space of the dual of L.
• More generally, given a vector bundle (finite-rank locally free sheaf) E on X, if R=Sym(E^*) 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 be a coherent sheaf on a Deligne–Mumford stack X. Then let For any , since global Spec is a right adjoint to the direct image functor, we have: ; in particular, is a commutative group scheme over X.
• Let R be a graded -algebra such that and is coherent and locally generates R as -algebra. Then there is a closed immersion

:

given by . Because of this, is called the abelian hull of the cone For example, if for some ideal sheaf I, then this embedding is the embedding of the normal cone into the normal bundle.

Computations

Consider the complete intersection ideal and let be the projective scheme defined by the ideal sheaf . Then, we have the isomorphism of -algebras given by

Properties

If 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

In particular, assuming R₀ = O_X, 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 . The projective cone is called the projective completion of C_R. Indeed, the zero-locus t = 0 is exactly 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 R₀ = O_X and R is locally generated as O_X-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 x_i'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=Spec_XR 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.

Notes

References

Lecture Notes

References

• § 8 of