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).
• 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})).^[1] 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 given by^{[citation needed]}

\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 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: 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 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 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:

\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.

Notes

1. ^ Behrend & Fantechi 1997, § 1.

References

Lecture Notes

• Fantechi, Barbara, "An introduction to Intersection Theory"

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, Vol. 2, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2nd ed., Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4. MR 1644323
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