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Veronese embedding pulls O(1) back to O(d)

Statement

Assume the Axiom of Choice as inherited from the projective-space and sheaf constructions (The Axiom of Choice). Let $S$ be a scheme, let $n \geq 0$ and $d \geq 1$ , and let $M$ be the set of multi-indices $m = (m_{0}, \dots, m_{n})$ with $m_{j} \geq 0$ and $∣ m ∣ = m_{0} + \dots + m_{n} = d$ ; put $N = ∣ M ∣ - 1 = (\binom{n + d}{d}) - 1.$ Write $P_{S}^{N}$ for the relative projective space whose standard coordinates are indexed by $M$ (Relative projective space from standard charts), with coordinate sections $y_{m} \in Γ (P_{S}^{N}, O (1))$ on the target and $x_{j} \in Γ (P_{S}^{n}, O (1))$ on the source, and let $s_{m} = x^{m} = \prod_{j = 0}^{n} x_{j}^{m_{j}} \in Γ (P_{S}^{n}, O (d))$ be the degree- $d$ monomial sections (Relative very ampleness in the finite projective-space convention). Let $ν_{d} : P_{S}^{n} ⟶ P_{S}^{N}$ be the associated morphism.

Then the monomial sections $s_{m}$ generate $O_{P_{S}^{n}} (d)$ (Global generation by the evaluation map) and $ν_{d}$ is a closed immersion with $ν_{d}^{*} O_{P_{S}^{N}} (1) ≅ O_{P_{S}^{n}} (d),$ carrying the coordinate section $y_{m}$ to $s_{m}$ . For $d = 1$ one has $N = n$ and $ν_{1}$ is the identity morphism; for $n = 0$ the morphism $ν_{d}$ is an isomorphism $P_{S}^{0} ≅ P_{S}^{0}$ . All schemes may be empty and no Noetherian or field hypothesis is imposed.

Facts & Assumptions

Given: A scheme $S$ , integers $n \geq 0$ and $d \geq 1$ , the index set $M$ of degree- $d$ monomials with $∣ M ∣ = (\binom{n + d}{d})$ , the relative projective spaces $P_{S}^{n}$ and $P_{S}^{N}$ with their standard charts, and the Axiom of Choice as inherited from the projective-space and sheaf constructions.

[A1]

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

[F1]

The standard charts $U_{i}$ of $P_{S}^{n}$ are affine over $S$ , and over an affine base $T = Spec A \subseteq S$ the chart $U_{i}^{T} = U_{i} \times_{S} T$ is $Spec A [x_{ℓ}^{(i)} : ℓ \neq i]$ with $x_{ℓ}^{(i)} = t_{ℓ} / t_{i}$ ; the charts and their overlaps commute with base change. The twisting sheaf $O (1)$ is glued from frames $e_{i}$ on $U_{i}$ with $e_{j} = x_{j}^{(i)} e_{i}$ on overlaps, and the coordinate sections satisfy $x_{i} ∣_{U_{i}} = e_{i}$ and $x_{j} ∣_{U_{i}} = x_{j}^{(i)} e_{i}$ for $j \neq i$ , so that $X_{x_{i}} = D_{+} (x_{i}) = U_{i}$ . For $d \geq 1$ the sheaf $O (d) = O (1)^{\otimes d}$ has frame $e_{i}^{d}$ on $U_{i}$ , and the monomial sections restrict to $s_{m} ∣_{U_{i}} = (\prod_{ℓ \neq i} (x_{ℓ}^{(i)})^{m_{ℓ}}) e_{i}^{d}$ , so that on $U_{i}$ one has $X_{s_{m}} = D (\prod_{ℓ \neq i} (x_{ℓ}^{(i)})^{m_{ℓ}})$ . (Relative projective space from standard charts, Relative very ampleness in the finite projective-space convention, Affine n-space over an arbitrary base)

[F2]

Let $X$ be an $S$ -scheme, $L$ an invertible $O_{X}$ -module and $t_{m} \in Γ (X, L)$ , indexed by a finite set, global sections generating $L$ . Then there is a unique $S$ -morphism $φ : X \to P_{S}^{N}$ such that $φ^{*} O (1) ≅ L$ with $φ^{*} (y_{m})$ corresponding to $t_{m}$ under this isomorphism and $φ^{- 1} (D_{+} (y_{m})) = X_{t_{m}}$ ; moreover on the chart $D_{+} (y_{m_{0}})$ with coordinates $y_{m} / y_{m_{0}}$ one has $(y_{m} / y_{m_{0}}) \circ φ = t_{m} / t_{m_{0}}$ on $X_{t_{m_{0}}}$ . (Generating line-bundle sections define a morphism to projective space)

[F3]

If finitely many global sections $t_{0}, \dots, t_{r}$ of an invertible sheaf $L$ induce a surjective morphism $O_{X}^{r + 1} \to L$ , $(g_{0}, \dots, g_{r}) \mapsto \sum_{i} g_{i} t_{i}$ , then $L$ is globally generated. (Global generation by the evaluation map)

[F4]

For a commutative ring $B$ and an ideal $I \subseteq B$ the quotient map $B \to B / I$ induces a closed immersion $Spec (B / I) \to Spec B$ ; a surjective ring homomorphism $B \to C$ induces a closed immersion after identifying $C$ with $B / \ker$ . (Closed immersions are affine quotients and survive base change, Closed immersions of schemes)

[F5]

A morphism $i : Z \to X$ is a closed immersion if and only if its restrictions $i^{- 1} (V_{j}) \to V_{j}$ to the members of an open cover $X = ⋃_{j} V_{j}$ are closed immersions. (Closed immersions are local on the target)

[F6]

A morphism is an immersion when it factors as a closed immersion followed by an open immersion; for a factorization $f = j \circ c$ with $j$ an open immersion the image $f (Z) = c (Z)$ is locally closed, and if it is closed in $X$ then $f$ is a closed immersion. (Immersion of schemes, An immersion with closed image is a closed immersion)

[F7]

Assume AC. For every scheme $S$ and every $r \geq 0$ the projection $P_{S}^{r} \to S$ is proper; hence for an $S$ -morphism $h : X \to Y$ with $X \to S$ proper and $Y \to S$ separated the morphism $h$ is proper; a proper morphism is a closed map, so the image of the whole source is closed. (Finite-dimensional projective space is proper over every base, Morphisms from a proper scheme to a separated one are proper, Proper morphisms are closed)

Proof

technique · direct: prove that the monomial sections generate $\mathcal O(d)$, let the universal property produce $\nu_d$, check on affine base opens that each source chart maps to a target chart by a surjective ring map hence a closed immersion, use locality on the target to exhibit $\nu_d$ as an immersion, and use properness to close the image

1.1F1F3algebra

The monomial sections generate $O (d)$ . Fix a chart $U_{i}$ and let $m = i^{d}$ be the multi-index with $m_{i} = d$ ; by [F1] the restriction $s_{i^{d}} ∣_{U_{i}} = e_{i}^{d}$ is a frame of $O (d)$ on $U_{i}$ , so the component $O_{X} \to O (d)$ of the evaluation morphism indexed by $i^{d}$ is an isomorphism over $U_{i}$ . Hence the evaluation morphism $O_{X}^{∣ M ∣} \to O (d)$ , $(g_{m}) \mapsto \sum_{m} g_{m} s_{m}$ , restricts to a surjection on every $U_{i}$ ; the charts cover $P_{S}^{n}$ , so it is surjective, and the $s_{m}$ generate $O (d)$ by [F3].

2.1F1F2step 1.1

The morphism and its pullback identity. By step 1.1 the sections $s_{m}$ generate the invertible sheaf $O (d)$ , so [F2] applies with $X = P_{S}^{n}$ , $L = O (d)$ and $t_{m} = s_{m}$ : there is a unique $S$ -morphism $ν_{d} : P_{S}^{n} \to P_{S}^{N}$ with $ν_{d}^{*} O (1) ≅ O (d)$ carrying $y_{m}$ to $s_{m}$ , and $ν_{d}^{- 1} (D_{+} (y_{m})) = X_{s_{m}}$ . Write $V_{m} = D_{+} (y_{m}) \subseteq P_{S}^{N}$ for the target chart at $m$ ; taking $m = i^{d}$ , [F1] gives $X_{s_{i^{d}}} = X_{x_{i}^{d}} = X_{x_{i}} = U_{i}$ (here $d \geq 1$ ), so $ν_{d}^{- 1} (V_{i^{d}}) = U_{i}$ for every $i$ , and this proves the pullback identity asserted.

3.1F1F2F4step 2.1algebra

The chartwise ring map is surjective. Let $T = Spec A$ be an affine open of $S$ . By base change [F1] the source chart is $U_{i}^{T} = Spec A [x_{ℓ}^{(i)} : ℓ \neq i]$ and the target chart $V_{i^{d}}^{T} = V_{i^{d}} \times_{S} T$ has coordinate ring $A [u_{m} : m \in M, m \neq i^{d}]$ with $u_{m} = y_{m} / y_{i^{d}}$ . On $U_{i} = X_{s_{i^{d}}}$ the chart formula of [F2] gives $u_{m} \circ ν_{d} = s_{m} / s_{i^{d}}$ , and by [F1] this is $s_{m} / s_{i^{d}} = \prod_{ℓ \neq i} (x_{ℓ}^{(i)})^{m_{ℓ}}$ because $s_{m} ∣_{U_{i}} = (\prod_{ℓ \neq i} (x_{ℓ}^{(i)})^{m_{ℓ}}) e_{i}^{d}$ and $s_{i^{d}} ∣_{U_{i}} = e_{i}^{d}$ . Consequently the induced $A$ -algebra map $A [u_{m}] \to A [x_{ℓ}^{(i)}]$ sends $u_{i^{d - 1} ℓ}$ to $x_{ℓ}^{(i)}$ for each $ℓ \neq i$ , hence is surjective, and by [F4] the base-changed morphism $U_{i}^{T} \to V_{i^{d}}^{T}$ is a closed immersion.

4.1F5step 3.1

Each chart gives a closed immersion. Fix $i$ . The open subschemes $V_{i^{d}}^{T}$ , for affine opens $T \subseteq S$ , cover $V_{i^{d}}$ , and the restriction of $ν_{d} ∣_{U_{i}} : U_{i} \to V_{i^{d}}$ to $V_{i^{d}}^{T}$ is the base-changed morphism $U_{i}^{T} \to V_{i^{d}}^{T}$ of step 3.1, which is a closed immersion; by [F5] therefore $ν_{d} ∣_{U_{i}} : U_{i} \to V_{i^{d}}$ is a closed immersion.

5.1F5F6step 2.1step 4.1

$ν_{d}$ is an immersion. Let $W = ⋃_{i = 0}^{n} V_{i^{d}} \subseteq P_{S}^{N}$ , an open subscheme containing the image of $ν_{d}$ because $ν_{d} (U_{i}) \subseteq V_{i^{d}}$ by step 2.1. Let $j : W ↪ P_{S}^{N}$ be the open immersion and $ν^{'} : P_{S}^{n} \to W$ the morphism with $ν_{d} = j \circ ν^{'}$ . The opens $V_{i^{d}}$ cover $W$ and $(ν^{'})^{- 1} (V_{i^{d}}) = U_{i}$ , so all restrictions of $ν^{'}$ to this cover are the closed immersions of step 4.1; by [F5] the morphism $ν^{'}$ is a closed immersion, and then $ν_{d} = j \circ ν^{'}$ is an immersion by [F6].

6.1F6F7step 5.1

$ν_{d}$ is a closed immersion. Since $ν_{d}$ is an immersion by step 5.1 and is an $S$ -morphism, and since $P_{S}^{n} \to S$ is proper while $P_{S}^{N} \to S$ is proper hence separated, [F7] shows that $ν_{d}$ is proper; a proper morphism is closed, so the image $ν_{d} (P_{S}^{n})$ is closed in $P_{S}^{N}$ , and an immersion with closed image is a closed immersion by [F6].

7.1

Conclusion and degenerate cases. Steps 1.1, 2.1 and 6.1 together show that the $s_{m}$ generate $O (d)$ and that $ν_{d}$ is a closed immersion with $ν_{d}^{*} O (1) ≅ O (d)$ carrying $y_{m}$ to $s_{m}$ . If $d = 1$ then $M = {e_{0}, \dots, e_{n}}$ and the identity of $P_{S}^{n}$ has pullback data $(O (1); x_{0}, \dots, x_{n})$ , so $ν_{1} = i d$ by the uniqueness in [F2]. If $n = 0$ then $P_{S}^{0} ≅ S$ , $O (d) = O_{S}$ and the single monomial section $x_{0}^{d}$ is a frame, so $ν_{d} : P_{S}^{0} \to P_{S}^{0}$ is an $S$ -morphism of $S$ to itself and a closed immersion by step 6.1, hence an isomorphism; for $S = \emptyset$ all four projective spaces are empty and $ν_{d}$ is the unique isomorphism $\emptyset \to \emptyset$ , which is also a closed immersion. The Axiom of Choice [A1] is inherited from the projective-space constructions and the gluing data of [F1]; no further choice is made. [A1, F1, F2, step 6.1, cases: d=1 and n=0 and empty S] \qed

Depends on

Used by

The conic map from O(2) Example

Dependency tree · two levels

73 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

The Stacks Project, Constructions of Schemes, Sections 27.8-27.21 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, 29 August 2022, Sections 4.5, 7.4, 9.3, 10.6, 17.4, 17.6, 18.2 (standard reference, not scraped)
Gao-Zhang, Lectures on Algebraic Geometry, Chapter 5 (standard reference, not scraped)