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 OX-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 $$\mathbb{P}(E)$$.
 * Let $$\mathcal{F}$$ be a coherent sheaf on a Deligne–Mumford stack X. Then let $$C(\mathcal{F}) := \operatorname{Spec}_X(\operatorname{Sym}(\mathcal{F})).$$ For any $$f: T \to X$$, since global Spec is a right adjoint to the direct image functor, we have: $$C(\mathcal{F})(T) = \operatorname{Hom}_{\mathcal{O}_X}(\operatorname{Sym}(\mathcal{F}), f_* \mathcal{O}_T)$$; in particular, $$C(\mathcal{F})$$ is a commutative group scheme over X.
 * Let R be a graded $$\mathcal{O}_X$$-algebra such that $$R_0 = \mathcal{O}_X$$ and $$R_1$$ is coherent and locally generates R as $$R_0$$-algebra. Then there is a closed immersion
 * $$\operatorname{Spec}_X R \hookrightarrow C(R_1)$$
 * given by $$\operatorname{Sym}(R_1) \to R$$. Because of this, $$C(R_1)$$ is called the abelian hull of the cone $$\operatorname{Spec}_X R.$$ 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 $$(f,g_1,g_2,g_3) \subset \mathbb{C}[x_0,\ldots,x_n]$$ and let $$X$$ be the projective scheme defined by the ideal sheaf $$\mathcal{I} = (f)(g_1,g_2,g_3)$$. Then, we have the isomorphism of $$\mathcal{O}_{\mathbb{P}^n}$$-algebras is given by

\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)} $$

Properties
If $$S \to R$$ is a graded homomorphism of graded OX-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 = OX, 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: explicitly, the n-th degree piece is
 * $$R_n \oplus R_{n-1} t \oplus R_{n-2} t^2 \oplus \cdots \oplus R_0 t^n$$.

Then the affine cone of it is denoted by $$C_{R[t]} = C_R \oplus 1$$. The projective cone $$\mathbb{P}(C_R \oplus 1)$$ is called the projective completion of CR. Indeed, the zero-locus t = 0 is exactly $$\mathbb{P}(C_R)$$ and the complement is the open subscheme CR. The locus t = 0 is called the hyperplane at infinity.

O(1)
Let R be a quasi-coherent graded OX-algebra such that R0 = OX and R is locally generated as OX-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 xi'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=SpecXR 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.