Cone (algebraic Geometry)

In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme ''X'', the relative Spec
:$C = \operatorname_X R$
of a quasi-coherent graded ''O''_''X''-algebra ''R'' is called the cone or affine cone of ''R''. Similarly, the relative Proj
:$\mathbb(C) = \operatorname_X R$
is called the projective cone of ''C'' or ''R''.

Note: The cone comes with the $\mathbb_m$-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 $R = \bigoplus_0^\infty I^n/I^$ for some ideal sheaf ''I'', then $\operatorname_X R$ is the normal cone to the closed scheme determined by ''I''.
*If $R = \bigoplus_0^\infty L^$ for some line bundle ''L'', then $\operatorname_X R$ 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 $\operatorname_X R$ is the total space of ''E'', often written just as ''E'', and the projective cone $\operatorname_X R$ is the projective bundle of ''E'', which is written as $\mathbb(E)$.
*Let $\mathcal$ be a coherent sheaf on a Deligne–Mumford stack ''X''. Then let $C(\mathcal) := \operatorname_X(\operatorname(\mathcal)).$ For any $f: T \to X$, since global Spec is a right adjoint to the direct image functor, we have: $C(\mathcal)(T) = \operatorname_(\operatorname(\mathcal), f_* \mathcal_T)$; in particular, $C(\mathcal)$ is a commutative group scheme over ''X''.
*Let ''R'' be a graded $\mathcal_X$-algebra such that $R_0 = \mathcal_X$ and $R_1$ is coherent and locally generates ''R'' as $R_0$-algebra. Then there is a closed immersion
::$\operatorname_X R \hookrightarrow C(R_1)$
:given by $\operatorname(R_1) \to R$. Because of this, $C(R_1)$ is called the abelian hull of the cone $\operatorname_X R.$ For example, if $R = \oplus_0^ I^n/I^$ 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 _0,\ldots,x_n/math> and let $X$ be the projective scheme defined by the ideal sheaf $\mathcal = (f)(g_1,g_2,g_3)$. Then, we have the isomorphism of $\mathcal_$-algebras is given by
:$\bigoplus_ \frac \cong \frac$

Properties

If $S \to R$ is a graded homomorphism of graded ''O''_''X''-algebras, then one gets an induced morphism between the cones:
:$C_R = \operatorname_X R \to C_S = \operatorname_X S$.
If the homomorphism is surjective, then one gets closed immersions $C_R \hookrightarrow C_S,\, \mathbb(C_R) \hookrightarrow \mathbb(C_S).$

In particular, assuming ''R''₀ = ''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., $X \overset\to C_R \to X$ 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
:$R_n \oplus R_ t \oplus R_ t^2 \oplus \cdots \oplus R_0 t^n$.
Then the affine cone of it is denoted by $C_ = C_R \oplus 1$. The projective cone $\mathbb(C_R \oplus 1)$ is called the projective completion of ''C''_''R''. Indeed, the zero-locus ''t'' = 0 is exactly $\mathbb(C_R)$ 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 ''R''₀ = ''O''_''X'' and ''R'' is locally generated as ''O''_''X''-algebra by ''R''₁. Then, by definition, the projective cone of ''R'' is:
:$\mathbb(C) = \operatorname_X R = \varinjlim \operatorname(R(U))$
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,
:$\operatorname(R(U)) \hookrightarrow \mathbb^r \times U.$
Then $\operatorname(R(U))$ has the line bundle ''O''(1) given by the hyperplane bundle $\mathcal_(1)$ of $\mathbb^r$; gluing such local ''O''(1)'s, which agree locally, gives the line bundle ''O''(1) on $\mathbb(C)$.

For any integer ''n'', one also writes ''O''(''n'') for the ''n''-th tensor power of ''O''(1). If the cone ''C''=Spec_''X''''R'' 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

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Reference

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*§ 8 of {{EGA , book=II

Algebraic geometry