Cone (algebraic geometry)

Generalization of a vector bundle

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

C = Spec X ⁡ R {\displaystyle C=\operatorname {Spec} _{X}R}

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

P ( C ) = Proj X ⁡ R {\displaystyle \mathbb {P} (C)=\operatorname {Proj} _{X}R}

is called the projective cone of C or R.

Note: The cone comes with the  G m {\displaystyle \mathbb {G} _{m}} -action due to the grading of R; this action is a part of the data of a cone (whence the terminology).

## Examples

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* 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 ∞ I n / I n + 1 {\displaystyle R=\bigoplus _{0}^{\infty }I^{n}/I^{n+1}} for some ideal sheaf I, then  Spec X ⁡ R {\displaystyle \operatorname {Spec} _{X}R} is the normal cone to the closed scheme determined by I.
* If  R = ⨁ 0 ∞ L ⊗ n {\displaystyle R=\bigoplus _{0}^{\infty }L^{\otimes n}} for some line bundle L, then  Spec X ⁡ R {\displaystyle \operatorname {Spec} _{X}R} 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  Spec X ⁡ R {\displaystyle \operatorname {Spec} _{X}R} is the total space of E, often written just as E, and the projective cone  Proj X ⁡ R {\displaystyle \operatorname {Proj} _{X}R} is the projective bundle of E, which is written as  P ( E ) {\displaystyle \mathbb {P} (E)} .
* Let  F {\displaystyle {\mathcal {F}}} be a coherent sheaf on a Deligne–Mumford stack X. Then let  C ( F ) := Spec X ⁡ ( Sym ⁡ ( F ) ) . {\displaystyle C({\mathcal {F}}):=\operatorname {Spec} _{X}(\operatorname {Sym} ({\mathcal {F}})).} [1] For any  f : T → X {\displaystyle f:T\to X} , since global Spec is a right adjoint to the direct image functor, we have:  C ( F ) ( T ) = Hom O X ⁡ ( Sym ⁡ ( F ) , f ∗ O T ) {\displaystyle C({\mathcal {F}})(T)=\operatorname {Hom} _{{\mathcal {O}}_{X}}(\operatorname {Sym} ({\mathcal {F}}),f_{*}{\mathcal {O}}_{T})} ; in particular,  C ( F ) {\displaystyle C({\mathcal {F}})} is a commutative group scheme over X.
* Let R be a graded  O X {\displaystyle {\mathcal {O}}_{X}} -algebra such that  R 0 = O X {\displaystyle R_{0}={\mathcal {O}}_{X}} and  R 1 {\displaystyle R_{1}} is coherent and locally generates R as  R 0 {\displaystyle R_{0}} -algebra. Then there is a closed immersion

Spec X ⁡ R ↪ C ( R 1 ) {\displaystyle \operatorname {Spec} _{X}R\hookrightarrow C(R_{1})}
given by  Sym ⁡ ( R 1 ) → R {\displaystyle \operatorname {Sym} (R_{1})\to R} . Because of this,  C ( R 1 ) {\displaystyle C(R_{1})} is called the abelian hull of the cone  Spec X ⁡ R . {\displaystyle \operatorname {Spec} _{X}R.} For example, if  R = ⊕ 0 ∞ I n / I n + 1 {\displaystyle R=\oplus _{0}^{\infty }I^{n}/I^{n+1}} for some ideal sheaf I, then this embedding is the embedding of the normal cone into the normal bundle.

### Computations

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Consider the complete intersection ideal  ( f , g 1 , g 2 , g 3 ) ⊂ C [ x 0 , … , x n ] {\displaystyle (f,g_{1},g_{2},g_{3})\subset \mathbb {C} [x_{0},\ldots ,x_{n}]} and let  X {\displaystyle X} be the projective scheme defined by the ideal sheaf  I = ( f ) ( g 1 , g 2 , g 3 ) {\displaystyle {\mathcal {I}}=(f)(g_{1},g_{2},g_{3})} . Then, we have the isomorphism of  O P n {\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}} -algebras is given by[citation needed]

⨁ n ≥ 0 I n I n + 1 ≅ O X [ a , b , c ] ( g 2 a − g 1 b , g 3 a − g 1 c , g 3 b − g 2 c ) {\displaystyle \bigoplus _{n\geq 0}{\frac {{\mathcal {I}}^{n}}{{\mathcal {I}}^{n+1}}}\cong {\frac {{\mathcal {O}}_{X}[a,b,c]}{(g_{2}a-g_{1}b,g_{3}a-g_{1}c,g_{3}b-g_{2}c)}}}

## Properties

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If  S → R {\displaystyle S\to R} is a graded homomorphism of graded OX-algebras, then one gets an induced morphism between the cones:

C R = Spec X ⁡ R → C S = Spec X ⁡ S {\displaystyle C_{R}=\operatorname {Spec} _{X}R\to C_{S}=\operatorname {Spec} _{X}S} .

If the homomorphism is surjective, then one gets closed immersions  C R ↪ C S , P ( C R ) ↪ P ( C S ) . {\displaystyle C_{R}\hookrightarrow C_{S},\,\mathbb {P} (C_{R})\hookrightarrow \mathbb {P} (C_{S}).}

In particular, assuming R0 = OX, the construction applies to the projection  R = R 0 ⊕ R 1 ⊕ ⋯ → R 0 {\displaystyle R=R_{0}\oplus R_{1}\oplus \cdots \to R_{0}} (which is an augmentation map) and gives

σ : X ↪ C R {\displaystyle \sigma :X\hookrightarrow C_{R}} .

It is a section; i.e.,  X → σ C R → X {\displaystyle X{\overset {\sigma }{\to }}C_{R}\to 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

R n ⊕ R n − 1 t ⊕ R n − 2 t 2 ⊕ ⋯ ⊕ R 0 t n {\displaystyle R_{n}\oplus R_{n-1}t\oplus R_{n-2}t^{2}\oplus \cdots \oplus R_{0}t^{n}} .

Then the affine cone of it is denoted by  C R [ t ] = C R ⊕ 1 {\displaystyle C_{R[t]}=C_{R}\oplus 1} . The projective cone  P ( C R ⊕ 1 ) {\displaystyle \mathbb {P} (C_{R}\oplus 1)} is called the projective completion of CR. Indeed, the zero-locus t = 0 is exactly  P ( C R ) {\displaystyle \mathbb {P} (C_{R})} and the complement is the open subscheme CR. The locus t = 0 is called the hyperplane at infinity.

## O(1)

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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 ) = Proj X ⁡ R = lim → ⁡ Proj ⁡ ( R ( U ) ) {\displaystyle \mathbb {P} (C)=\operatorname {Proj} _{X}R=\varinjlim \operatorname {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 ) ) ↪ P r × U . {\displaystyle \operatorname {Proj} (R(U))\hookrightarrow \mathbb {P} ^{r}\times U.}

Then  Proj ⁡ ( R ( U ) ) {\displaystyle \operatorname {Proj} (R(U))} has the line bundle O(1) given by the hyperplane bundle O P r ( 1 ) {\displaystyle {\mathcal {O}}_{\mathbb {P} ^{r}}(1)} of  P r {\displaystyle \mathbb {P} ^{r}} ; gluing such local O(1)'s, which agree locally, gives the line bundle O(1) on  P ( C ) {\displaystyle \mathbb {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

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1. ^ Behrend & Fantechi 1997, § 1.

## References

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### Lecture Notes

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

### References

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* Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1): 45–88\. arXiv:alg-geom/9601010. Bibcode:1997InMat.128...45B. doi:10.1007/s002220050136. ISSN 0020-9910.
* William Fulton. (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 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.