# 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 [1] 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 is 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 *R*_{0} = *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)

*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 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_{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.

## Notes

- Behrend–Fantechi, § 1.

## References

### Lecture Notes

- Fantechi, Barbara,
*An introduction to Intersection Theory*(PDF)

### Reference

- Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone".
*Inventiones Mathematicae*.**128**(1): 45–88. doi:10.1007/s002220050136. ISSN 0020-9910. - William Fulton. (1998),
*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.,**2**(2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323 - § 8 of Grothendieck, Alexandre; Dieudonné, Jean (1961). "Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes".
*Publications Mathématiques de l'IHÉS*.**8**. doi:10.1007/bf02699291. MR 0217084.