Cone (algebraic geometry)

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

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

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

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

is called the projective cone of C or R.

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

Examples

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

${\displaystyle \operatorname {Spec} _{X}R\hookrightarrow C(R_{1})}$

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

Computations

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

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

Properties

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

${\displaystyle C_{R}=\operatorname {Spec} _{X}R\to C_{S}=\operatorname {Spec} _{X}S}$.

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

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

${\displaystyle \sigma :X\hookrightarrow C_{R}}$.

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

${\displaystyle R_{n}\oplus R_{n-1}t\oplus R_{n-2}t^{2}\oplus \cdots \oplus R_{0}t^{n}}$.

Then the affine cone of it is denoted by ${\displaystyle C_{R[t]}=C_{R}\oplus 1}$. The projective cone ${\displaystyle \mathbb {P} (C_{R}\oplus 1)}$ is called the projective completion of C_R. Indeed, the zero-locus t = 0 is exactly ${\displaystyle \mathbb {P} (C_{R})}$ 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:

${\displaystyle \mathbb {P} (C)=\operatorname {Proj} _{X}R=\varinjlim \operatorname {Proj} (R(U))}$

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

${\displaystyle \operatorname {Proj} (R(U))\hookrightarrow \mathbb {P} ^{r}\times U.}$

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

For any integer n, one also writes O(n) for the n-th tensor power of O(1). If the cone C=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

1. Behrend–Fantechi, § 1. harvnb error: no target: CITEREFBehrend–Fantechi (help)

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.