Cone (algebraic geometry)

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

C

=

Spec

X

⁡

R

{\displaystyle C=\operatorname {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

{\displaystyle \mathbb {P} (C)=\operatorname {Proj} _{X}R}

is called the projective cone of C or R.

Note: The cone comes with the

G

m

{\displaystyle \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

=

⨁

0

∞

I

n

/

I

n

+

1

{\displaystyle R=\bigoplus _{0}^{\infty }I^{n}/I^{n+1}}

for some ideal sheaf I, then

Spec

X

⁡

R

{\displaystyle \operatorname {Spec} _{X}R}

is the normal cone to the closed scheme determined by I.

If

R

=

⨁

0

∞

L

⊗

n

{\displaystyle R=\bigoplus _{0}^{\infty }L^{\otimes n}}

for some line bundle L, then

Spec

X

⁡

R

{\displaystyle \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

Spec

X

⁡

R

{\displaystyle \operatorname {Spec} _{X}R}

is the total space of E, often written just as E, and the projective cone

Proj

X

⁡

R

{\displaystyle \operatorname {Proj} _{X}R}

is the projective bundle of E, which is written as

P

(

E

)

{\displaystyle \mathbb {P} (E)}

.

Let

F

{\displaystyle {\mathcal {F}}}

be a coherent sheaf on a Deligne–Mumford stack X. Then let

C

(

F

)

:=

Spec

X

⁡

(

Sym

⁡

(

F

)

)

.

{\displaystyle C({\mathcal {F}}):=\operatorname {Spec} _{X}(\operatorname {Sym} ({\mathcal {F}})).}

For any

f

:

T

→

X

{\displaystyle f:T\to X}

, since global Spec is a right adjoint to the direct image functor, we have:

C

(

F

)

(

T

)

=

Hom

O

X

⁡

(

Sym

⁡

(

F

)

,

f

∗

O

T

)

{\displaystyle C({\mathcal {F}})(T)=\operatorname {Hom} _{{\mathcal {O}}_{X}}(\operatorname {Sym} ({\mathcal {F}}),f_{*}{\mathcal {O}}_{T})}

; in particular,

C

(

F

)

{\displaystyle C({\mathcal {F}})}

is a commutative group scheme over X.

Let R be a graded

O

X

{\displaystyle {\mathcal {O}}_{X}}

-algebra such that

R

0

=

O

X

{\displaystyle R_{0}={\mathcal {O}}_{X}}

and

R

1

{\displaystyle R_{1}}

is coherent and locally generates R as

R

0

{\displaystyle R_{0}}

-algebra. Then there is a closed immersion

Spec

X

⁡

R

↪

C

(

R

1

)

{\displaystyle \operatorname {Spec} _{X}R\hookrightarrow C(R_{1})}

given by

Sym

⁡

(

R

1

)

→

R

{\displaystyle \operatorname {Sym} (R_{1})\to R}

. Because of this,

C

(

R

1

)

{\displaystyle C(R_{1})}

is called the abelian hull of the cone

Spec

X

⁡

R

.

{\displaystyle \operatorname {Spec} _{X}R.}

For example, if

R

=

⊕

0

∞

I

n

/

I

n

+

1

{\displaystyle 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

)

⊂

C

[

x

0

,

…

,

x

n

]

{\displaystyle (f,g_{1},g_{2},g_{3})\subset \mathbb {C} [x_{0},\ldots ,x_{n}]}

and let

X

{\displaystyle X}

be the projective scheme defined by the ideal sheaf

I

=

(

f

)

(

g

1

,

g

2

,

g

3

)

{\displaystyle {\mathcal {I}}=(f)(g_{1},g_{2},g_{3})}

. Then, we have the isomorphism of

O

P

n

{\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}}

-algebras given by

⨁

n

≥

0

I

n

I

n

+

1

≅

O

X

[

a

,

b

,

c

]

(

g

2

a

−

g

1

b

,

g

3

a

−

g

1

c

,

g

3

b

−

g

2

c

)

{\displaystyle \bigoplus _{n\geq 0}{\frac {{\mathcal {I}}^{n}}{{\mathcal {I}}^{n+1}}}\cong {\frac {{\mathcal {O}}_{X}[a,b,c]}{(g_{2}a-g_{1}b,g_{3}a-g_{1}c,g_{3}b-g_{2}c)}}}

Properties

If

S

→

R

{\displaystyle S\to 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

{\displaystyle C_{R}=\operatorname {Spec} _{X}R\to C_{S}=\operatorname {Spec} _{X}S}

.