Girish Mahajan (Editor)

Cone (algebraic geometry)

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In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, the relative Spec

Contents

C = Spec X ⁡ R

of a quasi-coherent graded OX-algebra R is called the cone or affine cone of R. Similarly, the relative Proj

P ( C ) = 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 = ⨁ 0 ∞ I n / I n + 1 for some ideal sheaf I, then Spec X ⁡ R is the normal cone to the closed scheme determined by I.
  • If R = ⨁ 0 ∞ L ⊗ n for some line bundle L, then 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 Spec X ⁡ R is the total space of E, often written just as E, and the projective cone Proj X ⁡ R is the projective bundle of E, which is written as P ( E ) .

    • Properties

    If S → R is a graded homomorphism of graded OX-algebras, then one gets an induced morphism between the cones:

    C R = Spec X ⁡ R → C S = Spec X ⁡ S .

    If the homomorphism is surjective, then one gets closed immersions C R ↪ C S , P ( C R ) ↪ P ( C S ) .

    In particular, assuming R0 = OX, the construction applies to the projection R = R 0 ⊕ R 1 ⊕ ⋯ → R 0 (which is an augmentation map) and gives

    σ : X ↪ C R .

    It is a section; i.e., X → σ C R → X is the identity and is called the zero-section embedding.

    Consider the graded algebra R[t] with variable t having degree one. 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 CR. Indeed, the zero-locus t = 0 is exactly 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 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:

    P ( C ) = Proj X ⁡ R = lim → ⁡ 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,

    Proj ⁡ ( R ( U ) ) ↪ P r × U .

    Then Proj ⁡ ( R ( U ) ) has the line bundle O(1) given by the hyperplane bundle 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=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.

    References

    Cone (algebraic geometry) Wikipedia
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