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Global generation does not imply very ampleness

Counterexample

Let $k$ be a field and let $X = P_{k}^{1} = Proj k [x_{0}, x_{1}]$ with its structure sheaf $O_{X}$ (Projective space is Proj of a polynomial ring). Then:

$O_{X}$ is globally generated by its unit section $1 \in Γ (X, O_{X})$ (Global generation by the evaluation map);
$O_{X}$ is not H-very ample over $Spec k$ (Relative very ampleness in the finite projective-space convention): the datum $(O_{X}; 1)$ with its single generating section $1$ has $n = 0$ and corresponds to the constant structure morphism $X \to P_{k}^{0} = Spec k$ , and for no $n \geq 0$ is there a quasi-compact $Spec k$ -immersion $i : X \to P_{k}^{n}$ with $i^{*} O (1) ≅ O_{X}$ .

So global generation of an invertible sheaf does not imply relative very ampleness, even over a field. This distinguishes global generation from relative very ampleness.

Facts & Assumptions

Given: A field $k$ , the scheme $X = P_{k}^{1} = Proj S$ with $S = k [x_{0}, x_{1}]$ graded by total degree, the standard opens $D_{+} (x_{0})$ , $D_{+} (x_{1})$ and the coordinate $t = x_{1} / x_{0}$ on $D_{+} (x_{0})$ , the structure sheaf $O_{X} = \tilde{S}$ , 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]

For homogeneous $f \in S_{+}$ of positive degree the standard open $D_{+} (f)$ is the affine chart $Spec S_{(f)}$ of $X$ ; the charts $D_{+} (x_{0})$ and $D_{+} (x_{1})$ cover $X$ with overlap $D_{+} (x_{0} x_{1})$ ; and $Γ (D_{+} (f), O_{X} (n)) = S (n)_{(f)}$ , with restrictions induced by homogeneous localisation, so that $Γ (D_{+} (f), O_{X}) = S_{(f)}$ for the structure sheaf. (Standard opens of Proj, Proj carries a scheme structure, Twisting sheaf on Proj, Sections of a graded-module sheaf on a standard open)

[F2]

A section of a sheaf on $X$ is the same as a compatible family of sections on the members of an open cover, and compatible local sections glue uniquely; the charts $U_{0} = D_{+} (x_{0})$ and $U_{1} = D_{+} (x_{1})$ are affine, hence quasi-compact, so their union $X$ is quasi-compact, and the structure morphism $X \to Spec k$ is quasi-compact. (A sheaf on a topological space, Every affine scheme is quasi-compact, A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact, Quasi-compact and quasi-separated schemes, Quasi-compact and quasi-separated morphisms)

[F3]

The structure sheaf $O_{X}$ is invertible; for a section $s$ of an invertible sheaf the nonvanishing locus $X_{s}$ is the set of points at which the image of $s$ in the fibre is nonzero; and the evaluation morphism of $O_{X}$ with its unit section is the canonical identification. (Invertible sheaves, Absolute ampleness by affine section opens, Global generation by the evaluation map)

[F4]

$P_{k}^{n}$ has standard charts $U_{i} = Spec k [x_{ℓ}^{(i)} : ℓ \neq i]$ , and $O (1)$ is glued from frames $e_{i}$ on $U_{i}$ by $e_{j} = x_{j}^{(i)} e_{i}$ , with coordinate sections $x_{j} \in Γ (P_{k}^{n}, O (1))$ restricting to $x_{j}^{(i)} e_{i}$ on $U_{i}$ and to $e_{j}$ on $U_{j}$ ; also $P_{k}^{0} = Spec k$ . A line bundle $L$ on $X$ is H-very ample relative to $Spec k$ exactly when there are an integer $n \geq 0$ and a quasi-compact $Spec k$ -immersion $i : X \to P_{k}^{n}$ with $i^{*} O (1) ≅ L$ . (Relative projective space from standard charts, Relative very ampleness in the finite projective-space convention)

[F5]

For every $S$ -scheme $Y$ the assignment $φ \mapsto (φ^{*} O (1); φ^{*} x_{0}, \dots, φ^{*} x_{n})$ is a natural bijection from $S$ -morphisms $Y \to P_{S}^{n}$ to isomorphism classes of pairs $(L; s_{0}, \dots, s_{n})$ with $L$ invertible and $s_{0}, \dots, s_{n}$ generating $L$ ; the $S$ -morphism $φ$ attached to such data satisfies $φ^{- 1} (D_{+} (x_{i})) = X_{s_{i}}$ and $x_{j}^{(i)} \circ φ = s_{j} / s_{i}$ on $X_{s_{i}}$ , the quotient being a regular function because $s_{i}$ trivialises $L$ there. (Maps to projective space equal generating line-bundle data, Generating line-bundle sections define a morphism to projective space)

[F6]

For a scheme $Y$ and a ring $A$ , taking global sections is a bijection $Hom (Y, Spec A) \to {Hom}_{C R i n g} (A, Γ (Y, O_{Y}))$ , so a morphism into an affine scheme is determined by its ring map on global sections; a morphism into the affine space $A_{S}^{n} = {Spec}_{O_{S}} O_{S} [t_{1}, \dots, t_{n}]$ is given by its $n$ coordinate functions. (Morphisms to an affine scheme and global sections, Affine n-space over an arbitrary base, Relative projective space from standard charts)

[F7]

A closed immersion is a homeomorphism onto a closed subset and an open immersion identifies its source with an open subscheme, so both are injective on points; an immersion is a composite of a closed immersion into an open subscheme followed by the inclusion of that open subscheme, hence is injective on points. A morphism that factors through a one-point scheme is constant on points, so it is not injective whenever its source has at least two points. (Closed immersions of schemes, Open immersions of schemes, Immersion of schemes)

[F8]

The polynomial ring $k [t]$ over the field $k$ is a domain, and $(0)$ and $(t)$ are two distinct prime ideals of $k [t]$ ; since $U_{0} ≅ Spec k [t]$ is an open subscheme of $X$ , the space $X$ has at least two points. (A polynomial ring over an integral domain is an integral domain, The underlying space of an affine spectrum, Projective space is Proj of a polynomial ring)

Refutation

technique · direct: compute the global sections of the structure sheaf by gluing its sections over the two standard charts and find that all of them are constants; then show that any morphism to projective space whose generating data consist of these constant sections factors through the structure morphism to the one-point scheme $\operatorname{Spec}k$, hence is constant, while an immersion is injective

1.1F1algebra

The chart rings. Every element of $S_{(x_{0})} = (S [x_{0}^{- 1}])_{0}$ is a class $a / x_{0}^{m}$ with $a \in S_{m}$ homogeneous of degree $m$ , and division by $x_{0}$ rewrites it as a polynomial in $t = x_{1} / x_{0}$ ; conversely every polynomial in $t$ arises this way. Hence $S_{(x_{0})} = k [t]$ , and symmetrically $S_{(x_{1})} = k [t^{- 1}]$ , so by [F1] one has $Γ (U_{0}, O_{X}) = k [t]$ and $Γ (U_{1}, O_{X}) = k [t^{- 1}]$ .

2.1F1step 1.1algebra

The overlap. The intersection $U_{0} \cap U_{1} = D_{+} (x_{0} x_{1})$ has $S_{(x_{0} x_{1})} = k [t, t^{- 1}]$ , and by [F1] the restriction maps are the homogeneous localisations $Γ (U_{0}, O_{X}) \to Γ (U_{0} \cap U_{1}, O_{X})$ and $Γ (U_{1}, O_{X}) \to Γ (U_{0} \cap U_{1}, O_{X})$ , that is, the inclusions $k [t] ↪ k [t, t^{- 1}]$ and $k [t^{- 1}] ↪ k [t, t^{- 1}]$ .

3.1F1F2step 1.1step 2.1algebra

Global sections of the structure sheaf. By [F2] a global section of $O_{X}$ is exactly a pair $(a, b)$ with $a \in k [t]$ , $b \in k [t^{- 1}]$ whose images in $k [t, t^{- 1}]$ agree, that is, $a (t) = b (t^{- 1})$ . Every element of $k [t, t^{- 1}]$ is a finite sum $\sum_{j} α_{j} t^{j}$ with $α_{j} \in k$ ; the left side involves only powers $j \geq 0$ and the right side only powers $j \leq 0$ , so all $α_{j}$ with $j \neq 0$ vanish and $a = b = α_{0} \in k$ . Hence $Γ (X, O_{X}) = k$ , consisting of the constant global functions.

4.1F3step 3.1algebra

The unit section generates. The unit section $1 \in Γ (X, O_{X}) = k$ is nonzero, and the evaluation morphism $Γ (X, O_{X}) \otimes_{Z} O_{X} \to O_{X}$ sends $1 \otimes g$ to $g \cdot 1 = g$ on every open $U \subseteq X$ , so it is surjective and $O_{X}$ is globally generated by $1$ . With [F3] it follows that $X_{1} = X$ , and more generally $X_{c \cdot 1} = X$ for $c \in k^{\times}$ while $X_{0 \cdot 1} = \emptyset$ , since a constant section has the same nonzero or zero value in every fibre.

5.1F2F4F5step 3.1step 4.1

A hypothetical immersion and its data. Suppose $O_{X}$ were H-very ample over $Spec k$ . By [F4] there are $n \geq 0$ and a quasi-compact $Spec k$ -immersion $i : X \to P_{k}^{n}$ with $i^{*} O (1) ≅ O_{X}$ ; the structure morphism $X \to Spec k$ is quasi-compact by [F2], so the definition applies. By [F5] the morphism $i$ is the one attached to the data $(i^{*} O (1); s_{0}, \dots, s_{n})$ with $s_{j} = i^{*} x_{j}$ , and these sections generate $i^{*} O (1)$ . Under the isomorphism $i^{*} O (1) ≅ O_{X}$ and the identification $Γ (X, O_{X}) = k$ of step 3.1 the sections $s_{j}$ correspond to constants $c_{j} \cdot 1$ with $c_{j} \in k$ , and not all $c_{j}$ vanish, since the $c_{j} \cdot 1$ generate the nonzero sheaf $O_{X}$ on the nonempty scheme $X$ ; fix $j$ with $c_{j} \neq 0$ .

6.1F4F5step 4.1step 5.1

The image lies in one affine chart. By [F5] the morphism $i$ satisfies $i^{- 1} (D_{+} (x_{j})) = X_{s_{j}}$ , and under the isomorphism $i^{*} O (1) ≅ O_{X}$ the nonvanishing locus $X_{s_{j}}$ is the nonvanishing locus of $c_{j} \cdot 1$ , which is all of $X$ by step 4.1 because $c_{j} \neq 0$ . Hence $i^{- 1} (D_{+} (x_{j})) = X$ : the image of $i$ is contained in the single standard chart $U_{j} = D_{+} (x_{j}) = Spec k [x_{ℓ}^{(j)} : ℓ \neq j]$ .

7.1F5step 5.1step 6.1algebra

The chart coordinates are constant. Again by [F5], on $X_{s_{j}} = X$ one has $x_{ℓ}^{(j)} \circ i = s_{ℓ} / s_{j}$ for every $ℓ \neq j$ ; under the identifications of step 5.1 this quotient is $(c_{ℓ} \cdot 1) / (c_{j} \cdot 1) = c_{ℓ} / c_{j} \in k$ , a constant regular function on $X$ .

8.1F6step 6.1step 7.1

The morphism is constant. Write $R_{j} = k [x_{ℓ}^{(j)} : ℓ \neq j]$ , so that $U_{j} = Spec R_{j}$ ; since $i$ maps into $U_{j}$ , it factors as $ι \circ i^{'}$ with $i^{'} : X \to U_{j}$ and $ι : U_{j} ↪ P_{k}^{n}$ the open immersion. By [F6] the morphism $i^{'}$ is determined by the ring map $(i^{'})^{#} : R_{j} \to Γ (X, O_{X}) = k$ , which sends $x_{ℓ}^{(j)}$ to the global function $x_{ℓ}^{(j)} \circ i^{'} = c_{ℓ} / c_{j}$ of step 7.1. This ring map is the composite of the ring homomorphism $R_{j} \to k$ , $x_{ℓ}^{(j)} \mapsto c_{ℓ} / c_{j}$ , with the structure map $k \to Γ (X, O_{X}) = k$ ; let $d : Spec k \to U_{j}$ be the morphism corresponding to $R_{j} \to k$ , $x_{ℓ}^{(j)} \mapsto c_{ℓ} / c_{j}$ under the bijection of [F6], and let $p : X \to Spec k$ be the structure morphism. Then $(d \circ p)^{#} = (i^{'})^{#}$ , so $i^{'} = d \circ p$ and $i = ι \circ d \circ p$ factors through the one-point scheme $Spec k$ .

9.1F7F8step 5.1step 8.1

Contradiction. By step 8.1 the morphism $i$ is constant on points, its image being the single point $d (Spec k)$ ; but by [F7] the immersion $i$ is injective on points, and by [F8] the source $X$ has at least two points. A constant map from a set with at least two points into any set is not injective, so no such $i$ exists for any $n \geq 0$ , and $O_{X}$ is not H-very ample over $Spec k$ .

10.1

Conclusion. The structure sheaf $O_{X}$ is globally generated by its unit section by step 4.1, while steps 5.1 to 9.1 show that it is not H-very ample over $Spec k$ . For $n = 0$ the same computation reads: the data $(O_{X}; 1)$ consist of a single generating section and correspond by [F5] to the morphism $X \to P_{k}^{0} = Spec k$ , which is the structure morphism and is constant, so the associated map to $P_{k}^{0}$ is constant and is not an immersion. Thus global generation does not imply relative very ampleness; $O_{X}$ is generated by one global section but is not H-very ample. The Axiom of Choice [A1] is inherited through the projective-space and data-equivalence suppliers; the only further data used are the two charts and the finite list of constants, so no choice is made here. [A1, F5, step 4.1, step 9.1, cases: n=0 and n at least 1] \qed

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Sources

The Stacks Project, Properties of Schemes, Section 28.27 (Tag 01PS) and Morphisms of Schemes, Definition 29.38.1 (Tag 01VG) (standard reference, not scraped)
Ravi Vakil, The Rising Sea, 29 August 2022, Sections 17.4, 17.6 and 18.2 (standard reference, not scraped)