Cone (algebraic geometry)
is called the projective cone of C or R.
Note: The cone comes with the -action due to the grading of R; this action is a part of the data of a cone (whence the terminology).
- 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 for some ideal sheaf I, then is the normal cone to the closed scheme determined by I.
- If for some line bundle L, then 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 is the total space of E, often written just as E, and the projective cone is the projective bundle of E, which is written as .
- Let be a coherent sheaf on a Deligne–Mumford stack X. Then let For any , since global Spec is a right adjoint to the direct image functor, we have: ; in particular, is a commutative group scheme over X.
- Let R be a graded -algebra such that and is coherent and locally generates R as -algebra. Then there is a closed immersion
- given by . Because of this, is called the abelian hull of the cone For example, if for some ideal sheaf I, then this embedding is the embedding of the normal cone into the normal bundle.
Consider the complete intersection ideal and let be the projective scheme defined by the ideal sheaf . Then, we have the isomorphism of -algebras is given by
If is a graded homomorphism of graded OX-algebras, then one gets an induced morphism between the cones:
If the homomorphism is surjective, then one gets closed immersions
In particular, assuming R0 = OX, the construction applies to the projection (which is an augmentation map) and gives
It is a section; i.e., 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
Then the affine cone of it is denoted by . The projective cone is called the projective completion of CR. Indeed, the zero-locus t = 0 is exactly and the complement is the open subscheme CR. The locus t = 0 is called the hyperplane at infinity.
Let R be a quasi-coherent 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:
where the colimit runs over open affine subsets U of X. By assumption R(U) has finitely many degree-one generators xi's. Thus,
Then has the line bundle O(1) given by the hyperplane bundle of ; gluing such local O(1)'s, which agree locally, gives the line bundle O(1) on .
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.
- Behrend–Fantechi, § 1.
- Fantechi, Barbara, An introduction to Intersection Theory (PDF)
- Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1): 45–88. doi:10.1007/s002220050136. ISSN 0020-9910.
- William Fulton. (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323
- § 8 of Grothendieck, Alexandre; Dieudonné, Jean (1961). "Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS. 8. doi:10.1007/bf02699291. MR 0217084.