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The twisting sheaf of projective space is very ample and ample

Statement

Assume the Axiom of Choice as inherited from the projective-space suppliers. Let $k$ be a field and let $n \geq 0$ , and write $P_{k}^{n} = P_{Spec k}^{n}$ for the relative projective space with its standard charts and twisting sheaf $O (1)$ (Relative projective space from standard charts, Relative very ampleness in the finite projective-space convention). Then $O_{P_{k}^{n}} (1)$ is closed H-very ample relative to $Spec k$ : the identity morphism of $P_{k}^{n}$ is a closed immersion over $Spec k$ and pulls $O (1)$ back to a sheaf isomorphic to $O (1)$ . Consequently

$O (1)$ is ample on $P_{k}^{n}$ (Absolute ampleness by affine section opens);
for every integer $d \geq 1$ the tensor power $O (d) = O (1)^{\otimes d}$ is ample on $P_{k}^{n}$ ;
for every finite morphism $g : Y \to P_{k}^{n}$ the pullback $g^{*} O (1)$ is an ample invertible $O_{Y}$ -module.

Facts & Assumptions

Given: a field $k$ , an integer $n \geq 0$ , the relative projective space $P_{k}^{n}$ over $Spec k$ with standard charts $U_{0}, \dots, U_{n}$ and twisting sheaf $O (1)$ .

[F1]

On relative projective space the charts $U_{i}$ are affine over the base and form an open cover, $O (1)$ is the invertible sheaf glued from free rank-one modules with transition $e_{i} \mapsto x_{i}^{(j)} e_{j}$ on $U_{i} \cap U_{j}$ (so that $e_{i}$ is a frame on $U_{i}$ ), and for $d \geq 0$ one sets $O (d) = O (1)^{\otimes d}$ , with $O (0) = O$ ; for $S = Spec A$ each chart is $Spec A [x_{ℓ}^{(i)}]$ and $P_{S}^{0} ≅ S$ . An invertible $O_{X}$ -module $L$ is H-very ample relative to $S$ when there are $m \geq 0$ and a quasi-compact $S$ -immersion $i : X \to P_{S}^{m}$ with $L ≅ i^{*} O (1)$ ; it is closed H-very ample when $i$ can be chosen a closed immersion (Relative projective space from standard charts, Relative very ampleness in the finite projective-space convention, Invertible sheaves).

[F2]

A morphism $i : Z \to X$ is a closed immersion when its underlying map is a homeomorphism onto a closed subset and the morphism $O_{X} \to i_{*} O_{Z}$ is surjective (Closed immersions of schemes).

[F3]

For a morphism of ringed spaces $f : X \to Y$ and an $O_{Y}$ -module $G$ the pullback is $f^{*} G = O_{X} \otimes_{f^{- 1} O_{Y}} f^{- 1} G$ , and for the identity morphism $f = {id}_{X}$ one has $f^{- 1} O_{X} = O_{X}$ and $f^{- 1} G = G$ ; the tensor product of an $O_{X}$ -module with the structure sheaf is canonically that module (Pullback of a module along a morphism of ringed spaces, Tensor product of sheaves of modules).

[F4]

If $f : X \to S$ is quasi-compact and $L$ is H-very ample relative to $S$ , then $L$ is $f$ -ample; if $S = Spec R$ is affine, $L$ is ample in the absolute sense (Relative very ampleness implies relative ampleness, Absolute ampleness by affine section opens).

[F5]

For a scheme $X$ , an invertible $O_{X}$ -module $L$ is ample if and only if $L^{\otimes m}$ is ample, for every integer $m \geq 1$ (Ampleness is invariant under positive powers).

[F6]

For a finite morphism $g : Y \to X$ and an ample invertible $O_{X}$ -module $L$ the pullback $g^{*} L$ is an ample invertible $O_{Y}$ -module, and if $X = \emptyset$ then $Y = \emptyset$ and $g^{*} L$ is the unique invertible sheaf on the empty scheme, which is ample (Finite pullback preserves absolute ampleness, Finite morphisms of schemes).

[F7]

The Axiom of Choice: every family of nonempty sets has a choice function (The Axiom of Choice).

Proof

Proof technique: direct; exhibit the identity morphism as the witnessing closed immersion, then apply the very-ample implication, power stability and finite pullback lemmas.

1.1F1F2given

(The identity witnesses closed H-very ampleness.) Take $m = n$ and $i = {id}_{P_{k}^{n}}$ in the definition [F1]: the identity is a morphism over $Spec k$ , and it is an isomorphism of schemes, hence a quasi-compact immersion of $P_{k}^{n}$ into itself; it is a closed immersion by the criterion [F2], because its underlying map is a homeomorphism of $P_{k}^{n}$ onto the closed subset $P_{k}^{n}$ and the morphism $O_{P_{k}^{n}} \to (id)_{*} O_{P_{k}^{n}}$ is the identity of the structure sheaf, which is surjective.

2.1F1F3step 1.1algebra

(The pullback identification.) For $f = {id}_{P_{k}^{n}}$ fact [F3] gives $f^{*} O (1) = O \otimes_{f^{- 1} O} f^{- 1} O (1)$ , and along the identity $f^{- 1} O = O$ and $f^{- 1} O (1) = O (1)$ ; the tensor product of the $O$ -module $O (1)$ with the structure sheaf is canonically $O (1)$ by [F3], so $f^{*} O (1) ≅ O (1)$ , and with step 1.1 this exhibits $O (1)$ as closed H-very ample relative to $Spec k$ in the sense of [F1].

3.1F1F4step 2.1

(Ampleness.) The structure morphism $P_{k}^{n} \to Spec k$ is quasi-compact because the finitely many affine charts $U_{0}, \dots, U_{n}$ of [F1] cover $P_{k}^{n}$ and each is affine, and the base $Spec k$ is affine, so [F4] applies to the H-very ample sheaf $O (1)$ of step 2.1 and gives that $O (1)$ is ample on $P_{k}^{n}$ in the absolute sense of [F1]; this is assertion 1.

4.1F1F5step 3.1algebra

(Positive powers.) Let $d \geq 1$ . By [F1] the sheaf $O (d)$ is $O (1)^{\otimes d}$ and is invertible, and by step 3.1 the sheaf $O (1)$ is ample, so applying [F5] with $m = d$ gives that $O (d)$ is ample; this is assertion 2, and its endpoint $d = 1$ is assertion 1 again.

4.2F6step 3.1

(Finite pullback.) Let $g : Y \to P_{k}^{n}$ be a finite morphism. By step 3.1 the sheaf $O (1)$ is ample, so [F6] gives that $g^{*} O (1)$ is an ample invertible $O_{Y}$ -module, and in the empty case $Y = \emptyset$ the same fact [F6] supplies the unique invertible sheaf of the empty scheme, which is ample; this is assertion 3.

5.1F1F7step 1.1step 4.1step 4.2∎

(Conclusion.) Assertions 1, 2 and 3 are established, and every use of choice above is inherited from the projective-space, very-ampleness and finite-morphism suppliers cited in [F1], [F4], [F5] and [F6] through the Axiom of Choice [F7]; the endpoints $n = 0$ and $d = 1$ are included, with $P_{k}^{0} ≅ Spec k$ and $O (1) ≅ O$ by [F1].

Depends on

Used by

Every smooth proper curve admits a projective embedding Corollary

Dependency tree · two levels

45 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ravi Vakil, The Rising Sea (version of October 21, 2025) (standard reference, not scraped)
The Stacks Project, Algebraic Curves (tag 0BRV) (standard reference, not scraped)