Cone (algebraic geometry) explained

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 ProjP (C) = \operatorname{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 = o plus ^{inf ty} _{0} 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 = o plus ^{inf ty} _{0} L ^{⊗} 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 P (E) .

• Let

l{F} be a coherent sheaf on a Deligne–Mumford stack X. Then let C( l{F}) : = \operatorname{Spec} _{X(\operatorname{Sym}(l{F})).} For any f: T \to X , since global Spec is a right adjoint to the direct image functor, we have: C( l{F})(T) = \operatorname{Hom} _{l{O} _{X}(\operatorname{Sym}(l{F}),f} _{*} l{O} _{T)} ; in particular, C( l{F}) is a commutative group scheme over X.

• Let R be a graded

l{O} _{X} -algebra such that R _{0} = l{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 = ⊕ ^{inf ty} _{0} 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,\ldots,x} _{n]} and let X be the projective scheme defined by the ideal sheaf l{I} = (f)(g _{1,g} _{2,g} _{3)} . Then, we have the isomorphism of l{O} _{P ^{n}} -algebras is given byo plus _{n\geq} (l{I)/(^{n}{l{I}}) ^{n+1} } \cong \frac

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,} P (C _{R)} \hookrightarrow P (C _{S).}

In particular, assuming R₀ = O_X, the construction applies to the projection

R = R _{0} ⊕ R _{1} ⊕ … \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} ⊕ R _{n-1} t ⊕ R _{n-2} t ^{2} ⊕ … ⊕ R _{0} t ^{n} .Then the affine cone of it is denoted by C _{R[t]} = C _{R} ⊕ 1 . The projective cone P (C _{R} ⊕ 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) = \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 P ^{r} x U. Then \operatorname{Proj}(R(U)) has the line bundle O(1) given by the hyperplane bundle l{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.

References

References

• Behrend. K.. Fantechi. B.. 1997-03-01. The intrinsic normal cone. Inventiones Mathematicae. en. 128. 1. 45–88. 10.1007/s002220050136. 0020-9910.
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