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 quasi-coherent 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 C or R.

Note: The cone comes with the $\mathbb {G} _{m}$ -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 $R=\bigoplus _{0}^{\infty }I^{n}/I^{n+1}$ 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, which agree locally, 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). 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.

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.

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