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Degree 2g does not force very ampleness

Statement refuted

Assume the Axiom of Choice; it supplies Dependent Choice through AC implies DC implies countable choice. The theorem that every line bundle of degree at least $2 g + 1$ on a curve of genus $g$ is very ample is sharp over any field $k$ when the curve has a $k$ -rational point: for a smooth proper geometrically integral curve $C / k$ of genus $g \geq 1$ with $p \in C (k)$ , the line bundle $O_{C} (K_{C} + 2 p)$ has degree $2 g$ and need not be very ample. When $k$ is algebraically closed, every closed point is $k$ -rational, so this includes the algebraically closed-field case.

Facts & Assumptions

Given: the Axiom of Choice and its consequence Dependent Choice; a field $k$ , a smooth proper geometrically integral curve $C$ of genus $g \geq 1$ over $k$ , a $k$ -rational point $p \in C (k)$ , a canonical divisor $K_{C}$ , and the invertible sheaf $L = O_{C} (K_{C} + 2 p)$ .

[F1]

On an integral proper curve over a field, divisors are finite integral sums of closed points and $\deg_{k} D = \sum_{x} n_{x} [κ (x) : k]$ ; since the given point $p$ is $k$ -rational, $\deg_{k} (p) = 1$ . Moreover $\deg_{k} (K_{C}) = 2 g - 2$ for every choice of the nonzero rational differential defining $K_{C}$ , the sheaf $ω_{C} = Ω_{C / k}^{1}$ satisfies $ω_{C} ≅ O_{C} (K_{C})$ , and under the divisor--invertible-sheaf dictionary $L (- p) = O_{C} (K_{C} + p)$ and $L (- 2 p) = O_{C} (K_{C})$ , with the corresponding inclusions of spaces of global sections. (Degree divisor proper curve, The canonical divisor has degree 2g - 2, Canonical bundle and canonical divisors, Invertible sheaf of cartier divisor, Cartier and Weil divisors agree on a smooth curve)

[F2]

For a divisor $D$ of degree $> 2 g - 2$ on a smooth proper geometrically integral curve of genus $g$ the nonspecial formula holds: $ℓ (D) = \deg_{k} D + 1 - g$ and $H^{1} (C, O_{C} (D)) = 0$ , equivalently $h^{0} (C, O_{C} (D)) = \deg_{k} D + 1 - g$ . (Riemann-Roch in exact form for divisors of degree above 2g - 2, H^1 of a line bundle vanishes above degree 2g - 2, The full Riemann-Roch theorem for divisors on a smooth proper curve, The Riemann-Roch dimension l(D))

[F3]

$ℓ (K_{C}) = h^{0} (C, ω_{C}) = g$ for any canonical divisor $K_{C}$ . (The canonical bundle has exactly g independent sections, The Riemann-Roch dimension l(D))

[F4]

A closed point $q$ is a base point of the complete linear system $∣ D ∣$ of a divisor $D$ exactly when $H^{0} (C, O_{C} (D - q)) = H^{0} (C, O_{C} (D))$ , equivalently when every global section of $O_{C} (D)$ vanishes at $q$ ; $D$ is base-point-free when it has no base point. A base-point-free subspace $V \subseteq L (D)$ of dimension $r + 1 \geq 1$ defines a morphism $φ_{V} : C \to P_{k}^{r}$ with $φ_{V}^{*} O (1) ≅ O_{C} (D)$ ; its system members are the pullbacks of hyperplanes. Conversely, if a morphism $i : C \to P_{k}^{n}$ is equipped with an isomorphism $i^{*} O (1) ≅ O_{C} (D)$ , its pulled-back coordinate sections form an ordered generating tuple, and the projective-data theorem reconstructs $i$ from that tuple. Their span is a base-point-free subspace of $L (D)$ , but may have dimension less than $n + 1$ ; equality with the basis morphism from the span up to a projective-linear change applies when the pulled-back coordinates are linearly independent. (Base points and base-point-free linear systems, Complete linear system, A base-point-free linear system defines a morphism to projective space, Maps to projective space equal generating line-bundle data)

[F5]

An invertible $O_{C}$ -module $L$ is closed $H$ -very ample relative to $Spec k$ when there is a closed immersion $i : C \to P_{k}^{n}$ with $i^{*} O (1) ≅ L$ ; $H$ -very ample relative to $Spec k$ means such an immersion exists that is quasi-compact. A locally closed immersion has injective differential at every point of its source (for a closed immersion this is the injectivity of the induced maps of Zariski tangent spaces, and an open immersion is an isomorphism onto its image); in particular an immersion $i : C \to P_{k}^{n}$ has $d i_{x} \neq 0$ at every closed point $x$ of $C$ , because the tangent space $T_{x} C$ of the smooth curve $C$ is one-dimensional. (Relative very ampleness in the finite projective-space convention)

[F6]

The very-ampleness theorem whose degree bound is tested here: if $\deg_{k} L \geq 2 g + 1$ , then $L$ is closed $H$ -very ample relative to $Spec k$ and the associated morphism $φ_{L}$ is a closed immersion. (Line bundles of degree at least 2g+1 are very ample)

[F7]

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

[F8]

In ZF, the Axiom of Choice implies Dependent Choice; this supplies the Dependent Choice premise of the Cartier-to-Weil divisor dictionary used in [F1] and the cited divisor suppliers. (AC implies DC implies countable choice, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain)

[F9]

On a genus-one curve, every canonical divisor satisfies $K_{C} \sim 0$ by The canonical bundle of a genus-one curve is trivial. A nonconstant morphism of smooth proper curves is finite, and the degree of a pulled-back line bundle is the map degree times its degree; on $P_{k}^{1}$ , $\deg O (1) = 1$ . Also $H^{0} (C, O_{C}) = k$ , so a constant $k$ -morphism from $C$ to $P_{k}^{1}$ has image a $k$ -rational point and pulls $O (1)$ back to a trivial line bundle. (Degree of a nonconstant morphism of curves, Fibres, pullbacks and degrees of divisors under a finite morphism of curves, Divisors on the projective line are classified by degree, Functions on a proper curve)

[F10]

Assuming AC as inherited from the duality suppliers, if $C$ is a smooth proper geometrically integral curve of genus $g$ over any field $k$ and an invertible sheaf $E$ has degree at least $2 g$ , then $E$ is base-point-free and its complete linear system defines a $k$ -morphism to $P_{k}^{h^{0} (C, E) - 1}$ pulling back $O (1)$ to $E$ . (Line bundles of degree at least 2g are base-point-free)

Counterexample

Proof technique: compute the image of the evaluation map on two-jets at $p$ ; it is one-dimensional, so no immersion with pullback $L$ can be nonzero on tangent spaces at $p$ .

1.1F1

Since $p$ is $k$ -rational, $\deg_{k} (p) = 1$ , so $\deg_{k} (K_{C} + 2 p) = (2 g - 2) + 2 = 2 g$ and $\deg_{k} (K_{C} + p) = 2 g - 1$ ; moreover $L (- p) = O_{C} (K_{C} + p)$ and $L (- 2 p) = O_{C} (K_{C}) = ω_{C}$ as subsheaves of the sheaf of rational sections.

1.2F4

Let $i : C \to P_{k}^{n}$ be any $k$ -morphism with an isomorphism $α : i^{*} O (1) \to L$ . Put $t_{j} = α (i^{*} x_{j})$ for the projective coordinates $x_{0}, \dots, x_{n}$ . The projective-data theorem in [F4] says that these sections form an ordered generating tuple and reconstruct $i$ from that tuple. Their span $W_{i} \subseteq H^{0} (C, L)$ is base-point-free, but its dimension may be less than $n + 1$ if the coordinate sections are linearly dependent.

1.3F7F8F1F2F4F6

The Axiom of Choice is used through the degree, duality, linear-system, and projective-data suppliers; [F8] supplies Dependent Choice for the Cartier-to-Weil divisor route.

2.1F4algebra

Local computation at $p$ : since the tuple in Step 1.2 generates $L$ , one of its coordinate sections is nonzero at $p$ ; after renumbering coordinates call it $t_{0}$ . Use $t_{0}$ as a local frame to identify $L_{p} / m_{p}^{2} L_{p}$ with $O_{C, p} / m_{p}^{2}$ , where the jet of $t_{0}$ is the class of $1$ . On the affine chart of $P_{k}^{n}$ where the corresponding coordinate is nonzero, $i$ is given by the ratios $t_{j} / t_{0}$ ; therefore $d i_{p} = 0$ exactly when every ratio has zero differential, equivalently when the jet of each $t_{j}$ lies in the line $k \cdot [t_{0}]$ . This says precisely that the image of $W_{i}$ in $L_{p} / m_{p}^{2} L_{p}$ is one-dimensional.

2.2F2step 1.1

Since $\deg_{k} L = 2 g > 2 g - 2$ , the divisor $D = K_{C} + 2 p$ is nonspecial and [F2] gives $h^{0} (C, L) = ℓ (D) = \deg_{k} D + 1 - g = g + 1$ together with $H^{1} (C, L) = 0$ .

2.3F1F2F3step 1.1

By Step 1.1, $L (- 2 p) = O_{C} (K_{C}) = ω_{C}$ and $L (- p) = O_{C} (K_{C} + p)$ ; hence $h^{0} (C, L (- 2 p)) = h^{0} (C, ω_{C}) = g$ by [F3], and $h^{0} (C, L (- p)) = g$ by [F2] applied to the divisor $K_{C} + p$ of degree $2 g - 1 > 2 g - 2$ .

2.4F10step 1.1

Since $\deg_{k} L = 2 g$ , theorem [F10] applies over the given arbitrary field $k$ and shows that $L$ is base-point-free. Thus the complete linear system $∣ L ∣$ is defined over $k$ .

3.1F2step 2.2step 2.3algebra

Let $A = O_{C, p}$ and $m = (t)$ , where $t$ is a uniformizer; choose a local frame $e$ of $L_{p}$ . Since $A / m = κ (p) = k$ and $m / m^{2}$ is one-dimensional over $k$ , the exact sequence $0 \to m / m^{2} \to A / m^{2} \to A / m \to 0$ gives $\dim_{k} (A / m^{2}) = 2$ . Hence $L_{p} / m^{2} L_{p} ≅ e (A / m^{2})$ is two-dimensional, with value and first-order classes represented by $e$ and $t e$ . The kernel of $e v : H^{0} (C, L) \to L_{p} / m^{2} L_{p}$ consists of sections whose stalk at $p$ lies in $m^{2} L_{p}$ ; since $L (- 2 p)$ has stalk $m^{2} L_{p}$ at $p$ and agrees with $L$ away from $p$ , this kernel is exactly $H^{0} (C, L (- 2 p))$ . Therefore $\dim_{k} im e v = h^{0} (C, L) - h^{0} (C, L (- 2 p)) = (g + 1) - g = 1$ .

3.2F4step 2.2step 2.4

By Step 2.4 the space $V = H^{0} (C, L)$ is base-point-free of dimension $g + 1$ , so [F4] provides the morphism $φ_{L} : C \to P_{k}^{g}$ of the complete linear system, with $φ_{L}^{*} O (1) ≅ L$ ; its target is $P_{k}^{h^{0} (C, L) - 1} = P_{k}^{g}$ .

4.1step 3.1step 1.2step 2.1

For the arbitrary morphism $i$ of Step 1.2, $W_{i} \subseteq H^{0} (C, L)$ , so its image in $L_{p} / m_{p}^{2} L_{p}$ is contained in the one-dimensional image of $H^{0} (C, L)$ from Step 3.1. It contains the nonzero jet of $t_{0}$ from Step 2.1, so its image is exactly one-dimensional and Step 2.1 gives $d i_{p} = 0$ .

5.1F5step 4.1

By [F5] an immersion has injective differential at every point and the tangent space $T_{p} C$ is one-dimensional, so $d i_{p} = 0$ means that $i$ is not an immersion at $p$ ; Steps 1.2--4.1 apply to every $k$ -morphism $i : C \to P_{k}^{n}$ with $i^{*} O (1) ≅ L$ and every $n$ . Thus no such morphism is a closed or locally closed immersion, and $L$ is not closed $H$ -very ample relative to $Spec k$ .

6.1F4step 3.2step 5.1

The morphism $φ_{L}$ of Step 3.2 is one of the morphisms covered by Step 5.1, so $d φ_{L} = 0$ at $p$ and $φ_{L}$ is not a closed immersion; its differential vanishes at the point $p$ of the base-point-free system $∣ L ∣$ .

7.1F1F3F4F9step 2.2step 6.1

For $g = 1$ , [F9] gives $K_{C} \sim 0$ (the chosen representative need not equal the zero divisor), hence $L ≅ O_{C} (2 p)$ . Step 2.2 gives $h^{0} (C, L) = 2$ , so the complete linear system morphism has target $P_{k}^{1}$ and pulls back $O (1)$ to $L$ . It is nonconstant because a constant map to a $k$ -rational point pulls $O (1)$ back to a trivial line bundle, whereas $\deg_{k} L = 2$ . By [F9], it is finite and $\deg_{k} L = \deg (φ_{L}) \deg O_{P^{1}} (1) = \deg (φ_{L})$ , so it is a morphism of degree two. The divisors in the pencil $∣ 2 p ∣$ are pullbacks of $k$ -rational points of $P_{k}^{1}$ and have degree two; a closed point $q$ pulls back to degree $2 [κ (q) : k]$ by [F9].

8.1F6step 5.1step 7.1∎

In summary $L$ has $\deg_{k} L = 2 g$ , is base-point-free by Step 2.4, and is not closed $H$ -very ample relative to $Spec k$ by Step 5.1; since [F6] gives closed $H$ -very ampleness for every line bundle of degree at least $2 g + 1$ , the hypothesis $\deg \geq 2 g + 1$ cannot be weakened to $\deg \geq 2 g$ and the bound $2 g + 1$ is sharp.

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Sources

William Fulton, Algebraic Curves (Internet Archive copy), Ch. 8 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (version of October 21, 2025) (standard reference, not scraped)
MIT 18.725 Algebraic Geometry (Fall 2015) course notes, Lectures 24-25 (standard reference, not scraped)