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 $C(\mathcal{F}) := \operatorname{Spec}_X(\operatorname{Sym}(\mathcal{F})).$ {{harvnb|Behrend|Fantechi|1997|loc=§ 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 is given by{{citation needed|date=August 2019}}

: $\bigoplus_{n\geq 0 } \frac{\mathcal{I}^n}{\mathcal{I}^{n+1}} \cong \frac{\mathcal{O}_X[a,b,c]}{(g_2a - g_1b, g_3a - g_1c, g_3b - g_2c)}$

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

References

= Lecture Notes =

{{Citation | last= Fantechi | first=Barbara | title=An introduction to Intersection Theory | url=http://www.cmi.ac.in/~asengupta/Intersection_Theory.pdf }}

= References =

{{Cite journal|last1=Behrend|first1=K.|last2=Fantechi|first2=B.|date=1997-03-01|title=The intrinsic normal cone|journal=Inventiones Mathematicae|language=en|volume=128|issue=1|pages=45–88|doi=10.1007/s002220050136|arxiv=alg-geom/9601010 |bibcode=1997InMat.128...45B |issn=0020-9910}}
{{Citation | title=Intersection theory | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-62046-4 | mr=1644323 | year=1998 | volume=2 | edition=2nd | author=William Fulton.}}
§ 8 of {{EGA | book=II}}

Category:Algebraic geometry