Cone (algebraic geometry)

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

$C = \operatorname{Spec}_X R$

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

$\mathbb{P}(C) = \operatorname{Proj}_X R$

is called the projective cone of R.

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 $R = \bigoplus_0^\infty I^n/I^{n+1}$ for some ideal sheaf I, then $\operatorname{Spec}_X R$ is the normal cone to the closed scheme determined by I.
• If $R = \bigoplus_0^\infty L^{\otimes n}$ for some line bundle L, then $\operatorname{Spec}_X R$ 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 $\operatorname{Spec}_X R$ is the total space of E, often written just as E, and the projective cone $\operatorname{Proj}_X R$ is the projective bundle of E, which is written as $\mathbb{P}(E)$.

Properties

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

$C_R = \operatorname{Spec}_X R \to C_S = \operatorname{Spec}_X S$.

If the homomorphism is surjective, then one gets closed immersions $C_R \hookrightarrow C_S,\, \mathbb{P}(C_R) \hookrightarrow \mathbb{P}(C_S).$

In particular, assuming R₀ = O_X, the construction applies to the projection $R = R_0 \oplus R_1 \oplus \cdots \to R_0$ (which is an augmentation map) and gives

$\sigma: X \hookrightarrow C_R$.

It is a section; i.e., $X \overset{\sigma}\to C_R \to X$ is the identity and is called the zero-section embedding.

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

$\mathbb{P}(C) = \operatorname{Proj}_X R = \varinjlim \operatorname{Proj}(R(U))$

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,

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

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

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

References

• 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.