Cone (algebraic geometry)

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

[math]\displaystyle{ C = \operatorname{Spec}_X R }[/math]

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

[math]\displaystyle{ \mathbb{P}(C) = \operatorname{Proj}_X R }[/math]

is called the projective cone of C or R.

Note: The cone comes with the [math]\displaystyle{ \mathbb{G}_m }[/math]-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 [math]\displaystyle{ R = \bigoplus_0^\infty I^n/I^{n+1} }[/math] for some ideal sheaf I, then [math]\displaystyle{ \operatorname{Spec}_X R }[/math] is the normal cone to the closed scheme determined by I.
If [math]\displaystyle{ R = \bigoplus_0^\infty L^{\otimes n} }[/math] for some line bundle L, then [math]\displaystyle{ \operatorname{Spec}_X R }[/math] 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 [math]\displaystyle{ \operatorname{Spec}_X R }[/math] is the total space of E, often written just as E, and the projective cone [math]\displaystyle{ \operatorname{Proj}_X R }[/math] is the projective bundle of E, which is written as [math]\displaystyle{ \mathbb{P}(E) }[/math].
Let [math]\displaystyle{ \mathcal{F} }[/math] be a coherent sheaf on a Deligne–Mumford stack X. Then let [math]\displaystyle{ C(\mathcal{F}) := \operatorname{Spec}_X(\operatorname{Sym}(\mathcal{F})). }[/math]^[1] For any [math]\displaystyle{ f: T \to X }[/math], since global Spec is a right adjoint to the direct image functor, we have: [math]\displaystyle{ C(\mathcal{F})(T) = \operatorname{Hom}_{\mathcal{O}_X}(\operatorname{Sym}(\mathcal{F}), f_* \mathcal{O}_T) }[/math]; in particular, [math]\displaystyle{ C(\mathcal{F}) }[/math] is a commutative group scheme over X.
Let R be a graded [math]\displaystyle{ \mathcal{O}_X }[/math]-algebra such that [math]\displaystyle{ R_0 = \mathcal{O}_X }[/math] and [math]\displaystyle{ R_1 }[/math] is coherent and locally generates R as [math]\displaystyle{ R_0 }[/math]-algebra. Then there is a closed immersion

[math]\displaystyle{ \operatorname{Spec}_X R \hookrightarrow C(R_1) }[/math]

given by [math]\displaystyle{ \operatorname{Sym}(R_1) \to R }[/math]. Because of this, [math]\displaystyle{ C(R_1) }[/math] is called the abelian hull of the cone [math]\displaystyle{ \operatorname{Spec}_X R. }[/math] For example, if [math]\displaystyle{ R = \oplus_0^{\infty} I^n/I^{n+1} }[/math] 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 [math]\displaystyle{ (f,g_1,g_2,g_3) \subset \mathbb{C}[x_0,\ldots,x_n] }[/math] and let [math]\displaystyle{ X }[/math] be the projective scheme defined by the ideal sheaf [math]\displaystyle{ \mathcal{I} = (f)(g_1,g_2,g_3) }[/math]. Then, we have the isomorphism of [math]\displaystyle{ \mathcal{O}_{\mathbb{P}^n} }[/math]-algebras is given by^{[citation needed]}

[math]\displaystyle{ \bigoplus_{n\geq 0 } \frac{\mathcal{I}^n}{\mathcal{I}^{n+1}} \cong \frac{\mathcal{O}_X[a,b,c]}{(g_2a - g_1b, g_3a - g_1c, g_3b - g_2c)} }[/math]

Properties

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

[math]\displaystyle{ C_R = \operatorname{Spec}_X R \to C_S = \operatorname{Spec}_X S }[/math].

If the homomorphism is surjective, then one gets closed immersions [math]\displaystyle{ C_R \hookrightarrow C_S,\, \mathbb{P}(C_R) \hookrightarrow \mathbb{P}(C_S). }[/math]

In particular, assuming R₀ = O_X, the construction applies to the projection [math]\displaystyle{ R = R_0 \oplus R_1 \oplus \cdots \to R_0 }[/math] (which is an augmentation map) and gives

[math]\displaystyle{ \sigma: X \hookrightarrow C_R }[/math].

It is a section; i.e., [math]\displaystyle{ X \overset{\sigma}\to C_R \to X }[/math] 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

[math]\displaystyle{ R_n \oplus R_{n-1} t \oplus R_{n-2} t^2 \oplus \cdots \oplus R_0 t^n }[/math].

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

[math]\displaystyle{ \mathbb{P}(C) = \operatorname{Proj}_X R = \varinjlim \operatorname{Proj}(R(U)) }[/math]

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,

[math]\displaystyle{ \operatorname{Proj}(R(U)) \hookrightarrow \mathbb{P}^r \times U. }[/math]

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

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

↑ 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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[1] Behrend & Fantechi 1997, § 1.

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