Cone (algebraic geometry)

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

C = {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

ℙ (C) = {Proj}_{X} R

is called the projective cone of C or R.

Note: The cone comes with the $𝔾_{m}$ -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 $R = ⨁_{0}^{\infty} I^{n} / I^{n + 1}$ for some ideal sheaf I, then ${Spec}_{X} R$ is the normal cone to the closed scheme determined by I.
If $R = ⨁_{0}^{\infty} L^{\otimes n}$ for some line bundle L, then ${Spec}_{X} R$ 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 ${Spec}_{X} R$ is the total space of E, often written just as E, and the projective cone ${Proj}_{X} R$ 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 $C (ℱ) := {Spec}_{X} (Sym (ℱ)) .$ For any $f : T \to X$ , since global Spec is a right adjoint to the direct image functor, we have: $C (ℱ) (T) = {Hom}_{𝒪_{X}} (Sym (ℱ), f_{*} 𝒪_{T})$ ; in particular, $C (ℱ)$ is a commutative group scheme over X.
Let R be a graded $𝒪_{X}$ -algebra such that $R_{0} = 𝒪_{X}$ and $R_{1}$ is coherent and locally generates R as $R_{0}$ -algebra. Then there is a closed immersion

{Spec}_{X} R ↪ C (R_{1})

given by

Sym (R_{1}) \to R

. Because of this,

C (R_{1})

is called the abelian hull of the cone

{Spec}_{X} R .

For example, if

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 $(f, g_{1}, g_{2}, g_{3}) \subset ℂ [x_{0}, \dots, x_{n}]$ and let $X$ be the projective scheme defined by the ideal sheaf $ℐ = (f) (g_{1}, g_{2}, g_{3})$ . Then, we have the isomorphism of $𝒪_{ℙ^{n}}$ -algebras given by^{[citation needed]}

⨁_{n \geq 0} \frac{ℐ^{n}}{ℐ^{n + 1}} ≅ \frac{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 $S \to R$ is a graded homomorphism of graded O_X-algebras, then one gets an induced morphism between the cones:

C_{R} = {Spec}_{X} R \to C_{S} = {Spec}_{X} S

.

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

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

σ : X ↪ C_{R}

.

It is a section; i.e., $X \overset{σ}{\to} C_{R} \to X$ 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} \oplus R_{n - 1} t \oplus R_{n - 2} t^{2} \oplus \dots \oplus R_{0} t^{n}

.

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

ℙ (C) = {Proj}_{X} R = \lim_{\to} 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,

Proj (R (U)) ↪ ℙ^{r} \times U .

Then $Proj (R (U))$ has the line bundle O(1) given by the hyperplane bundle $𝒪_{ℙ^{r}} (1)$ of $ℙ^{r}$ ; 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=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 (PDF)

References

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.
Fulton, William (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.

[1] Behrend & Fantechi 1997, § 1.