Cone (algebraic geometry)

In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, the relative Spec

C=SpecXR

of a quasi-coherent graded OX-algebra R is called the cone or affine cone of R. Similarly, the relative Proj

ℙ(C)=ProjXR

is called the projective cone of C or R.

Note: The cone comes with the 𝔾m-action due to the grading of R; this action is a part of the data of a cone (whence the terminology).

## Contents

* 1 Examples
* 1.1 Computations
* 2 Properties
* 3 O(1)
* 4 Notes
* 5 References
* 5.1 Lecture Notes
* 5.2 References

## 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 R=⨁0∞In/In+1 for some ideal sheaf I, then SpecXR is the normal cone to the closed scheme determined by I.
* If R=⨁0∞L⊗n for some line bundle L, then SpecXR 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 SpecXR is the total space of E, often written just as E, and the projective cone ProjXR is the projective bundle of E, which is written as ℙ(E).
* Let ℱ be a coherent sheaf on a Deligne–Mumford stack X. Then let C(ℱ):=SpecX(Sym⁡(ℱ)).[1] For any f:T->X, since global Spec is a right adjoint to the direct image functor, we have: C(ℱ)(T)=Hom𝒪X(Sym⁡(ℱ),f*𝒪T); in particular, C(ℱ) is a commutative group scheme over X.
* Let R be a graded 𝒪X-algebra such that R0=𝒪X and R1 is coherent and locally generates R as R0-algebra. Then there is a closed immersion

SpecXR↪C(R1)
given by Sym⁡(R1)->R. Because of this, C(R1) is called the abelian hull of the cone SpecXR. For example, if R=⊕0∞In/In+1 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 (f,g1,g2,g3)⊂ℂ[x0,…,xn] and let X be the projective scheme defined by the ideal sheaf ℐ=(f)(g1,g2,g3). Then, we have the isomorphism of 𝒪ℙn-algebras is given by[citation needed]

⨁n≥0ℐnℐn+1≅𝒪X[a,b,c](g2a−g1b,g3a−g1c,g3b−g2c)

## Properties

If S->R is a graded homomorphism of graded OX-algebras, then one gets an induced morphism between the cones:

CR=SpecXR->CS=SpecXS.

If the homomorphism is surjective, then one gets closed immersions CR↪CS,ℙ(CR)↪ℙ(CS).

In particular, assuming R0 = OX, the construction applies to the projection R=R0⊕R1⊕⋯->R0 (which is an augmentation map) and gives

σ:X↪CR.

It is a section; i.e., X->σCR->X 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

Rn⊕Rn−1t⊕Rn−2t2⊕⋯⊕R0tn.

Then the affine cone of it is denoted by CR[t]=CR⊕1. The projective cone ℙ(CR⊕1) is called the projective completion of CR. Indeed, the zero-locus t = 0 is exactly ℙ(CR) and the complement is the open subscheme CR. 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:

ℙ(C)=ProjXR=lim->Proj⁡(R(U))

where the colimit runs over open affine subsets U of X. By assumption R(U) has finitely many degree-one generators xi's. Thus,

Proj⁡(R(U))↪ℙr×U.

Then Proj⁡(R(U)) has the line bundle O(1) given by the hyperplane bundle 𝒪ℙr(1) of ℙr; gluing such local O(1)'s, which agree locally, gives the line bundle O(1) on ℙ(C).

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.

## Notes

1. ↑ Behrend & Fantechi 1997, § 1.

## References

### Lecture Notes

* Fantechi, Barbara, An introduction to Intersection Theory, http://www.cmi.ac.in/~asengupta/Intersection_Theory.pdf

### References

* Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone" (in en). 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
* § 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. http://www.numdam.org/item/PMIHES_1961__8__5_0.

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