04.05.05 · algebraic-geometry / divisors

Ample and very ample line bundle

shipped3 tiersLean: partial

Anchor (Master): Hartshorne §II–§III; Vakil; Lazarsfeld *Positivity in Algebraic Geometry* Vol I; Grothendieck EGA II

Intuition [Beginner]

A line bundle $L$ on a scheme $X$ is very ample if its global sections embed $X$ as a closed subscheme of projective space. It is ample if some tensor power $L^{n}$ is very ample. Ample bundles are the positively curved line bundles of algebraic geometry: they witness the projectivity of $X$ and produce its embeddings.

Why does this matter? Projective embeddings are the foundational tool for studying varieties: a projective embedding gives a Hilbert polynomial, a degree, an arithmetic genus, and a host of cohomological invariants computable from the embedding. A variety is projective (in the sense of admitting a closed embedding into some projective space) iff it carries an ample line bundle. So ampleness encodes projectivity in line-bundle language.

The technical version is best understood via the Cartan-Serre-Grothendieck criterion: a line bundle $L$ on a complete scheme is ample iff for every coherent sheaf $F$ , the higher cohomology of the twist $F (n) := F$ -tensored-with- $L^{n}$ vanishes (i.e., $H^{i}$ vanishes for $i \geq 1$ and $n$ large). Ampleness is thus a cohomological positivity condition. It generalises to nef, big, and effective cones in birational geometry, and is at the heart of the minimal model program.

Visual [Beginner]

A scheme embedded in projective space via global sections of a very ample line bundle; the sections trace out a closed subvariety inside $P^{N}$ .

Worked example [Beginner]

On the projective line $P_{k}^{1}$ , the line bundle $O (1)$ is very ample. Its global sections are the linear forms in two variables, spanned by $x_{0}, x_{1}$ . The map

P_{k}^{1} \to P_{k}^{1}, [a : b] \mapsto [x_{0} (a, b) : x_{1} (a, b)] = [a : b]

is the identity embedding — $P_{k}^{1}$ embeds into itself by $O (1)$ .

The line bundle $O (2)$ on $P_{k}^{1}$ has global sections the homogeneous quadratics: $x_{0}^{2}, x_{0} x_{1}, x_{1}^{2}$ . These three sections give a map $P_{k}^{1} \to P_{k}^{2}$ , $[a : b] \mapsto [a^{2} : ab : b^{2}]$ , the Veronese embedding of degree 2. Its image is the smooth conic $V (y_{0} y_{2} - y_{1}^{2}) \subset P_{k}^{2}$ .

For each $n \geq 1$ , $O (n)$ on $P_{k}^{1}$ is very ample, embedding $P_{k}^{1}$ as the rational normal curve of degree $n$ in $P_{k}^{n}$ . The line bundle $O (0) = O_{P_{k}^{1}}$ is ample-but-not-very-ample? No: $O_{P^{1}}$ is not even ample, since its powers all stay structure-sheaf and produce no embedding. Only $O (d)$ for $d \geq 1$ is ample on $P_{k}^{1}$ ; for $d \geq 1$ it is also very ample.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X$ be a scheme over a field $k$ (or more generally, over a Noetherian base $S$ ), and let $L$ be a line bundle on $X$ .

Very ample. $L$ is very ample (over $k$ ) if there exists a closed immersion $ι : X ↪ P_{k}^{N}$ for some $N$ such that $ι^{*} O_{P^{N}} (1) ≅ L$ . Equivalently, the natural map

φ_{L} : X \to P (H^{0} (X; L)^{\lor}), x \mapsto [evaluation of sections at x]

is a closed immersion. This requires that $L$ is globally generated (the sections $H^{0} (X; L)$ have no common zero, equivalently $φ_{L}$ is defined everywhere) and that $φ_{L}$ separates points and tangent vectors.

Ample. $L$ is ample if there exists a positive integer $n$ such that $L^{\otimes n}$ is very ample. Equivalently, $L$ is ample iff for every coherent sheaf $F$ on $X$ , there exists $n_{0}$ depending on $F$ such that $F \otimes L^{\otimes n}$ is globally generated for all $n \geq n_{0}$ .

Cartan-Serre-Grothendieck criterion. Let $X$ be a Noetherian scheme proper over a Noetherian base ring $A$ . A line bundle $L$ on $X$ is ample iff one of the following equivalent conditions holds:

(A1) For every coherent sheaf $F$ on $X$ , $F \otimes L^{\otimes n}$ is globally generated for $n ≫ 0$ .

(A2) For every coherent sheaf $F$ on $X$ and every $i \geq 1$ , $H^{i} (X; F \otimes L^{\otimes n}) = 0$ for $n ≫ 0$ .

(A3) Some tensor power $L^{\otimes n}$ is very ample (over $A$ ).

This trio of conditions — global generation, cohomological vanishing, and projective embedding — is the heart of ampleness.

Numerical (Nakai-Moishezon) criterion. Let $X$ be a complete (proper over $k$ ) scheme. A line bundle $L$ is ample iff for every irreducible closed subscheme $Y \subseteq X$ of dimension $r \geq 1$ ,

(L^{r} \cdot Y) := \int_{Y} c_{1} (L)^{r} > 0.

(Intersection number positive on every positive-dimensional subvariety.)

This says ampleness is a numerical condition — it depends only on the class of $L$ in $NS (X) \otimes Q$ , the rational Néron-Severi group.

Globally generated. $L$ is globally generated (or base-point-free) if the evaluation map

H^{0} (X; L) \otimes_{k} O_{X} \to L

is surjective. Equivalently, for every $x \in X$ , there is a global section $s$ with $s (x) \neq = 0$ .

Examples.

Projective space. $O (d)$ on $P_{k}^{n}$ is ample iff $d > 0$ . It is very ample iff $d \geq 1$ (Veronese embeddings).
Affine schemes. Affine schemes do not carry ample line bundles (they admit no closed embedding into projective space, since they are non-projective).
Smooth projective curves. On a curve $C$ of genus $g$ , a line bundle $L$ is ample iff $de g L > 0$ . It is very ample iff $de g L \geq 2 g + 1$ (Castelnuovo-Riemann-Roch criterion).
Abelian varieties. On an abelian variety $A$ , an ample line bundle exists iff $A$ is projective (always true for abelian varieties over fields). The classical theta divisor on a Jacobian is the canonical ample bundle.
Failure of ampleness. On $P_{k}^{1} \times P_{k}^{1}$ , the bundle $O (1, 0)$ (pullback from one factor) has structure-sheaf restriction to the fibres of the other projection, so $(O (1, 0)^{2} \cdot fibre) = 0$ , violating Nakai-Moishezon. It is not ample, only nef (numerically effective).

Ample cone. The set of ample classes in $NS (X) \otimes R$ forms an open convex cone, the ample cone $Amp (X)$ . Its closure is the nef cone, dual under the intersection pairing to the Mori cone of effective curve classes (Kleiman's criterion). The geometry of these cones drives the minimal model program.

Key theorem with proof [Intermediate+]

Theorem (Cartan-Serre-Grothendieck criterion for ampleness). Let $X$ be a Noetherian scheme proper over a Noetherian ring $A$ , and let $L$ be a line bundle on $X$ . The following are equivalent.

(i) Some tensor power $L^{\otimes n}$ is very ample (over $A$ ).

(ii) For every coherent sheaf $F$ on $X$ , $F \otimes L^{\otimes n}$ is globally generated for $n ≫ 0$ .

(iii) For every coherent sheaf $F$ on $X$ and every $i \geq 1$ , $H^{i} (X; F \otimes L^{\otimes n}) = 0$ for $n ≫ 0$ .

Proof. The key implications are (i) $\Rightarrow$ (iii) $\Rightarrow$ (ii) $\Rightarrow$ (i).

(i) $\Rightarrow$ (iii): Suppose $L^{\otimes m}$ is very ample, giving a closed immersion $ι : X ↪ P_{A}^{N}$ with $ι^{*} O (1) = L^{\otimes m}$ . Serre's projective vanishing theorem on $P_{A}^{N}$ : for any coherent sheaf $G$ on $P_{A}^{N}$ , $H^{i} (P_{A}^{N}; G \otimes O (d)) = 0$ for $i \geq 1$ and $d ≫ 0$ . Apply this to $G = ι_{*} F$ on $P_{A}^{N}$ :

H^{i} (X; F \otimes L^{\otimes m d}) = H^{i} (P_{A}^{N}; ι_{*} F \otimes O (d)) = 0

for $d ≫ 0$ . Splitting tensor powers $L^{\otimes n}$ as $L^{\otimes m d + r}$ with $0 \leq r < m$ and applying the bound to each shift covers all $n ≫ 0$ . So (iii) holds.

(iii) $\Rightarrow$ (ii): Given $F$ coherent, the global sections functor is exact for sheaves with $H^{1} = 0$ . For $n ≫ 0$ , $H^{1} (X; m_{x} \cdot F \otimes L^{\otimes n}) = 0$ for every closed point $x \in X$ (where $m_{x}$ is the ideal sheaf), so the evaluation map

H^{0} (X; F \otimes L^{\otimes n}) \to F \otimes L^{\otimes n} ∣_{x}

is surjective at $x$ . Thus $F \otimes L^{\otimes n}$ is globally generated near $x$ . Noetherian compactness gives finitely many $x_{i}$ exhausting $X$ ; taking $n$ large enough for all gives (ii). (A finer argument using vanishing at higher orders and the openness of generation handles uniformity.)

(ii) $\Rightarrow$ (i): Apply (ii) with $F = O_{X}$ . For $n \geq n_{0}$ , $L^{\otimes n}$ is globally generated. The map $φ_{n} : X \to P (H^{0} (X; L^{\otimes n})^{\lor})$ is then defined everywhere. For larger $n$ — applying (ii) to $F = I_{x} \otimes I_{y}$ for pairs of distinct closed points and to $F = I_{x}^{2}$ for tangent-vector separation — $φ_{n}$ separates points and tangent vectors, hence is a closed immersion. So $L^{\otimes n}$ is very ample for $n$ sufficiently large. $□$

This theorem is the foundational equivalence: ampleness equals projective embedding equals cohomological positivity equals eventual global generation. Each formulation is useful in different contexts; the Cartan-Serre-Grothendieck triad organises them.

Bridge. The construction here builds toward 04.05.02 (picard group), where the same data is developed in the next layer of the strand. The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Exercise 1 (easy, proof).

Show that $O (d)$ on $P_{k}^{n}$ is very ample for $d \geq 1$ .

Hint

Use the global sections of $O (d)$ — homogeneous polynomials of degree $d$ .

Answer

The global sections $H^{0} (P_{k}^{n}; O (d))$ form the space of homogeneous polynomials of degree $d$ in $x_{0}, \dots, x_{n}$ , of dimension $(n n + d)$ .

The induced map $φ_{d} : P_{k}^{n} \to P_{k}^{N}$ with $N = (n n + d) - 1$ is given by all degree- $d$ monomials. This is the $d$ -th Veronese embedding.

To verify it is a closed immersion: separation of points requires that for distinct $[a], [b] \in P_{k}^{n}$ , some monomial of degree $d$ takes different values — for $d \geq 1$ this is satisfied (the linear forms already separate, since two distinct points differ in at least one coordinate). Separation of tangent vectors requires injectivity on tangent spaces — also holds for $d \geq 1$ .

The image is a closed subvariety of $P_{k}^{N}$ cut out by the Veronese relations: the degree-2 polynomials in monomial-coordinates that vanish on the image, expressing relations like $y_{i + j} y_{k + l} = y_{i + k} y_{j + l}$ when $i + j = i + k$ and so on (the polynomial-ring relations).

For $d = 1$ , the Veronese is the identity. For $d \geq 2$ , the image is the Veronese variety of dimension $n$ and degree $d^{n}$ in $P_{k}^{N}$ . $□$

Exercise 3 (medium, proof).

Prove that a line bundle on a smooth projective curve $C$ of genus $g$ is very ample if its degree is $\geq 2 g + 1$ .

Hint

Use Riemann-Roch and the criterion that very ample = separates points and tangent vectors.

Answer

Let $L$ be a line bundle on $C$ with $de g L = d$ . By Riemann-Roch, $h^{0} (L) - h^{1} (L) = d - g + 1$ , and Serre duality gives $h^{1} (L) = h^{0} (ω_{C} \otimes L^{- 1})$ .

If $d \geq 2 g + 1 > 2 g - 2 = de g ω_{C}$ , then $de g (ω_{C} \otimes L^{- 1}) = 2 g - 2 - d < 0$ , so $h^{0} (ω_{C} \otimes L^{- 1}) = 0$ . Hence $h^{1} (L) = 0$ and $h^{0} (L) = d - g + 1$ .

Separation of points. For distinct points $p, q \in C$ , the divisor $D = p + q$ has $h^{0} (L (- D)) = h^{0} (L \otimes O (- p - q))$ , with $de g (L (- D)) = d - 2 \geq 2 g - 1$ . By Riemann-Roch and the same argument, $h^{0} (L (- D)) = d - 2 - g + 1 = d - g - 1$ . So the codimension of sections vanishing at both $p, q$ inside all sections is $h^{0} (L) - h^{0} (L (- D)) = 2$ , i.e., the evaluation map $H^{0} (L) \to L ∣_{p} \oplus L ∣_{q}$ is surjective, separating $p$ and $q$ .

Separation of tangent vectors. Similar: for $p \in C$ , the divisor $2 p$ has $h^{0} (L (- 2 p)) = d - 2 - g + 1 = d - g - 1$ , and the codimension is 2 again, giving surjectivity of $H^{0} (L) \to L ∣_{p} \oplus T_{p}^{*} C \otimes L ∣_{p}$ — separating tangent vectors at $p$ .

Both conditions hold for $d \geq 2 g + 1$ , so $φ_{L} : C \to P (H^{0} (L)^{\lor})$ is a closed immersion: $L$ is very ample. $□$

For $g = 0$ : $d \geq 1$ suffices (the rational normal curve in $P^{d}$ ). For $g = 1$ : $d \geq 3$ suffices; the elliptic-curve embedding by $O (3 p)$ . For $g \geq 2$ : the canonical bundle $ω_{C}$ has degree $2 g - 2$ , but $ω_{C}$ is not very ample for hyperelliptic curves (the canonical map is 2:1 onto a rational curve); for non-hyperelliptic $g \geq 3$ , $ω_{C}$ is very ample.

Exercise 4 (medium, numeric).

Compute the degree of the image of $P_{k}^{1}$ under the embedding by $O (d)$ in $P_{k}^{d}$ .

Hint

Degree of the image equals the self-intersection of the very ample class.

Answer

The degree of $φ_{d} (P_{k}^{1}) \subset P_{k}^{d}$ as a projective curve is $de g O (d) = d$ (intersection of $φ_{d} (P_{k}^{1})$ with a hyperplane in $P_{k}^{d}$ ).

For $d = 1$ : degree 1 (a line in $P_{k}^{1}$ ). For $d = 2$ : degree 2 (the smooth conic $V (y_{0} y_{2} - y_{1}^{2})$ in $P_{k}^{2}$ ). For $d = 3$ : degree 3 (the twisted cubic in $P_{k}^{3}$ ). General $d$ : the rational normal curve of degree $d$ in $P_{k}^{d}$ .

These rational normal curves are characterised among smooth curves of degree $d$ in $P_{k}^{d}$ as the smooth projective curves of minimal degree relative to ambient dimension: for any smooth nondegenerate curve $C \subset P_{k}^{N}$ , $de g C \geq N$ , with equality iff $C$ is a rational normal curve (and then it is $P^{1}$ embedded by a complete linear system of degree $N$ ).

Exercise 5 (medium, conceptual).

Why does the Nakai-Moishezon criterion characterise ampleness numerically?

Hint

Ample classes form an open cone in the Néron-Severi group; the criterion gives the cone via inequalities.

Answer

The Nakai-Moishezon criterion: on a complete scheme $X$ , $L$ is ample iff $(L^{r} \cdot Y) > 0$ for every irreducible closed subscheme $Y$ of dimension $r \geq 1$ .

This is numerical because the intersection number $(L^{r} \cdot Y)$ depends only on the numerical class of $L$ — the image of $L$ in $NS (X) := Pic (X) / \equiv$ (numerical equivalence). Numerically equivalent line bundles have identical intersection numbers with every subvariety.

So ampleness factors through $NS (X)$ : it depends only on the class $[L] \in NS (X) \otimes Q$ , not on the specific bundle.

Combined with Kleiman's criterion for nefness ( $L$ nef iff $(L \cdot C) \geq 0$ for every irreducible curve $C$ ), and the fact that ample = interior of nef, one obtains the ample cone in $NS (X) \otimes R$ as an open convex cone whose closure is the nef cone. The Mori cone of effective curve classes is dual.

This cone of curves picture is the foundation of the minimal model program: the geometry of $X$ is encoded in the structure of these convex cones in $NS (X) \otimes R$ , and the cone theorem (Mori, Kawamata) classifies the extremal rays.

Exercise 6 (hard, proof).

State and outline a proof of the Kodaira embedding theorem: a compact Kähler manifold is projective iff it carries a positive line bundle.

Hint

Positive line bundle = ample, by Kodaira vanishing and Cartan-Serre-Grothendieck.

Answer

Kodaira embedding theorem. A compact complex Kähler manifold $X$ admits a closed embedding into some $P^{N} (C)$ iff there exists a holomorphic line bundle $L$ on $X$ whose first Chern class $c_{1} (L) \in H^{1, 1} (X; R)$ is represented by a positive real $(1, 1)$ -form (Kähler form, equivalently $c_{1} (L)$ inside the Kähler cone).

Outline. A positive line bundle $L$ has, by Kodaira's vanishing theorem,

H^{i} (X; L^{n} \otimes ω_{X}) = 0 for i \geq 1, n ≫ 0

(equivalently $H^{i} (X; L^{n}) = 0$ for $i \geq 1$ and $n \geq 1$ when $L$ is positive enough). This is the analytic version of the Cartan-Serre-Grothendieck cohomological criterion.

By the Cartan-Serre-Grothendieck criterion (in its analytic GAGA form), positivity of $L$ implies that $L^{n}$ is globally generated and separates points and tangent vectors for $n$ large. So $L^{n}$ is very ample: the map $φ_{n} : X \to P (H^{0} (X; L^{n})^{\lor})$ is a closed analytic embedding.

Conversely, a projective embedding $ι : X ↪ P^{N}$ pulls back the Fubini-Study form $ω_{F S}$ — a positive $(1, 1)$ -form representing $c_{1} (O (1))$ — to a positive form on $X$ representing $c_{1} (ι^{*} O (1))$ . So $X$ has a positive line bundle.

This theorem links three different perspectives:

Algebraic geometry: ample line bundle.
Differential geometry: positive curvature on a holomorphic line bundle.
Topology: projective embedding (compactness + closed in $P^{N}$ ).

Combined with GAGA (Serre 1956), Kodaira embedding shows: smooth compact Kähler manifolds with a positive line bundle = smooth complex projective varieties (analytically). The equivalence of the algebraic and analytic perspectives in this setting is one of the deepest results of mid-20th-century mathematics. $□$

Exercise 7 (hard, conceptual).

Explain the role of ample line bundles in the minimal model program.

Hint

The MMP runs flips and divisorial contractions until the canonical bundle becomes nef.

Answer

The minimal model program (MMP, also Mori program) seeks to classify higher-dimensional algebraic varieties up to birational equivalence. The key player is the canonical bundle $K_{X} = ω_{X}$ (or its $Q$ -Cartier extension on mildly singular varieties).

A variety is a minimal model if $K_{X}$ is nef (numerically effective: $(K_{X} \cdot C) \geq 0$ for every irreducible curve $C$ ). It is a Mori fibre space if $K_{X}$ is anti-ample on the fibres of a structure morphism.

The MMP procedure. Start with a smooth projective $X$ . If $K_{X}$ is nef, $X$ is a minimal model and stop. If not, by the cone theorem (Mori-Kawamata) there is a $K_{X}$ -negative extremal ray $R$ in the Mori cone $\overline{NE} (X)$ , and an associated contraction morphism $φ_{R} : X \to Y$ . Three cases:

Divisorial contraction: $φ_{R}$ contracts a divisor; replace $X$ by $Y$ .
Small contraction: $φ_{R}$ contracts a subvariety of codimension $\geq 2$ ; replace $X$ by a flip $X^{+} \to Y$ (existence of flips is the BCHM theorem, 2010, by Birkar-Cascini-Hacon-McKernan).
Mori fibre space: $dim Y < dim X$ ; stop with the Mori fibre space $X \to Y$ .

Iterate. Termination is conjectural in general (proved up to dimension 4) but the existence of flips (BCHM) and abundance (in low dimension) are major theorems.

Role of ample bundles. Ample classes are extremal points of the Kähler / nef cone. The MMP modifies $X$ to push the canonical class $K_{X}$ towards nef-ness, equivalently out of the anti-ample directions. The cone theorem identifies $K_{X}$ -negative extremal rays as exactly the directions to contract; ampleness of an auxiliary divisor $H + ϵ K_{X}$ (for $H$ ample, $ϵ > 0$ small) is what guarantees the contraction exists and is projective.

In summary: ampleness controls the projective and cohomological behaviour of the MMP, the cone of effective curves provides the dual data, and the canonical bundle is the central object whose positivity is the goal.

This perspective — varieties classified up to birational equivalence by the position of $K_{X}$ in the cone of divisors — is the major conceptual achievement of late-20th-century algebraic geometry and remains under active development (in characteristic $p$ , in the relative setting, for moduli of higher-dimensional varieties).

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has projective-space and invertible-sheaf API; ample/very ample definitions and the Cartan-Serre-Grothendieck criterion are partially formalised.

[object Promise]

Advanced results [Master]

Kodaira embedding theorem. A compact Kähler manifold $X$ is projective iff it carries a holomorphic line bundle whose first Chern class is positive. The proof uses Kodaira's vanishing theorem: a positive line bundle has all higher cohomology of $L^{n} \otimes ω_{X}$ vanishing for $n ≫ 0$ . By Cartan-Serre-Grothendieck (in its analytic form), this forces eventual very ampleness.

Kleiman's criterion. A line bundle $L$ on a complete scheme is nef (numerically effective) iff $(L \cdot C) \geq 0$ for every irreducible curve $C \subseteq X$ . The nef cone is the closure of the ample cone. Together they form the foundational pair of cones in $NS (X) \otimes R$ .

Seshadri's criterion. $L$ on a complete scheme is ample iff there exists $ϵ > 0$ such that for every closed point $x \in X$ and every irreducible curve $C$ through $x$ , $(L \cdot C) \geq ϵ \cdot mult_{x} (C)$ . The supremum of such $ϵ$ is the Seshadri constant $ϵ (L, x)$ , a refinement of ampleness measuring local positivity. Seshadri constants are major invariants in modern positivity theory (Lazarsfeld Positivity).

Big and effective cones. Beyond ample/nef: a line bundle $L$ is big if $h^{0} (L^{n}) \sim c \cdot n^{d i m X}$ for $c > 0$ . The big cone is open in $NS (X) \otimes R$ , and it contains the ample cone. The effective cone of classes admitting nonzero sections is even larger but typically not closed.

Iitaka dimension. For a line bundle $L$ , the Iitaka dimension $κ (L) := lim sup_{n \to \infty} lo g h^{0} (L^{n}) / lo g n$ takes values in ${- \infty, 0, 1, \dots, dim X}$ . $L$ ample $\Rightarrow κ = dim X$ (maximal); $L$ big $\Leftrightarrow κ = dim X$ .

Castelnuovo-Mumford regularity. A coherent sheaf $F$ on $P_{k}^{N}$ is $m$ -regular if $H^{i} (F (m - i)) = 0$ for $i \geq 1$ . Regularity bounds when $F (d)$ becomes globally generated and has higher cohomology vanishing — a quantitative refinement of ampleness statements. Regularity controls Hilbert-scheme constructions and resolution-of-singularities estimates.

Effective ampleness on abelian varieties. Lefschetz's theorem: on an abelian variety $A$ , a line bundle $L$ is very ample iff $L^{\otimes 3}$ has the property that $∣ L^{3} ∣$ separates points and tangent vectors. The Lefschetz-Mukai criterion and Mumford's index theorem on theta divisors give effective bounds.

Rationality and ampleness in characteristic $p$ . In positive characteristic, Kodaira vanishing can fail (Raynaud counterexamples), so Kodaira embedding requires modification. The Mehta-Ramanathan theorem and Frobenius-positivity substitutes (Hara, Smith, Schwede) provide replacement positivity criteria.

Asymptotic invariants. Volumes of line bundles $vol (L) := lim sup n^{- d i m X} \cdot n! \cdot h^{0} (L^{n})$ are continuous invariants on $NS (X) \otimes R$ (Boucksom-Demailly-Paun-Peternell), giving a real-analytic refinement of ampleness.

Synthesis. This construction generalises the pattern fixed in 04.05.03 (line bundle on a scheme), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Detailed proofs of: Cartan-Serre-Grothendieck (sketched in the formal section), Nakai-Moishezon (via reduction to curves and Riemann-Roch on each subvariety), Kleiman's criterion (via duality and the cone theorem), Kodaira embedding (Hodge-theoretic, using Kodaira vanishing), and the BCHM existence-of-flips theorem (uses positivity heavily) — these are deferred to companion units in the Hodge theory and minimal model program strands. The basic Cartan-Serre-Grothendieck triad is proved in the formal-definition section.

Connections [Master]

Line bundle 04.05.03 — ampleness is a property of line bundles; the ample cone sits inside $Pic (X) \otimes R$ .
Cartier divisor 04.05.04 — ampleness extends to Cartier divisors and $Q$ -Cartier divisors in birational geometry.
Picard group 04.05.02 — the ample cone is the natural positivity structure inside $Pic (X) \otimes R$ .
Coherent sheaf 04.06.02 — the Cartan-Serre-Grothendieck criterion is a coherent-sheaf property.
Sheaf cohomology 04.03.01 — Serre's vanishing on projective space and Kodaira vanishing are foundational ample-bundle cohomology theorems.
Projective space 04.07.01 — $O (d)$ on $P^{n}$ is the canonical example of an ample (very ample for $d \geq 1$ ) bundle.
Riemann-Roch theorem for curves 04.04.01 — Riemann-Roch and Castelnuovo's bound classify very ample bundles on curves by degree relative to genus.

Historical & philosophical context [Master]

The notion of ampleness emerged from the post-war French school's reformulation of projective geometry in sheaf-theoretic language. Jean-Pierre Serre's 1955 Faisceaux Algébriques Cohérents (FAC) introduced the twisting sheaves $O (d)$ on $P^{n}$ and proved the foundational vanishing theorem: for any coherent sheaf $F$ on $P^{n}$ , $H^{i} (P^{n}; F (d)) = 0$ for $i \geq 1$ and $d ≫ 0$ . This was the prototypical example of cohomological positivity, and it implied that $F (d)$ is globally generated for $d ≫ 0$ . Serre's twisting sheaves were the first ample line bundles, and FAC implicitly identified the cohomological-positivity property that defines ampleness.

Alexander Grothendieck and Jean Dieudonné, in Éléments de Géométrie Algébrique II (EGA II, 1961), developed the systematic theory of ample and very ample line bundles in scheme theory. EGA II §4 gives the formal definitions; §5 proves the Cartan-Serre-Grothendieck equivalence (often credited to Grothendieck, generalising Serre's projective-space case). Grothendieck's approach was characteristically functorial: ampleness is a property of a relative line bundle that controls the projectivity of a morphism, and the equivalence with cohomological vanishing is the natural way to articulate it within his sheaf-theoretic framework. EGA II also introduced the notion of ampleness relative to a base — a line bundle $L$ on $X \to S$ ample relative to $S$ — which is the right notion in moduli theory and arithmetic geometry.

Yozo Nakai (1963) and Boris Moishezon (1964) gave the numerical criterion: ampleness is detected by intersection numbers on subvarieties. The Nakai-Moishezon theorem reduced ampleness to a geometric inequality — a positivity condition on the class of the line bundle in the Néron-Severi group — and connected ampleness to the nascent theory of cones of divisors and curves.

Kunihiko Kodaira's 1953 vanishing theorem and the resulting Kodaira embedding theorem (also 1953) established the analytic counterpart: a compact Kähler manifold is projective iff it has a positive holomorphic line bundle. Kodaira's proof used differential-geometric methods (the Bochner technique on harmonic forms) and provided the bridge between algebraic and analytic perspectives via GAGA (Serre 1956). The Kodaira embedding theorem remains one of the deepest theorems in 20th-century complex analysis and algebraic geometry.

In contemporary research, ample line bundles are central to the minimal model program (Mori, Kawamata, Reid, Shokurov; modern work of Birkar-Cascini-Hacon-McKernan, Birkar 2016 BCHM-style finiteness). The ample, nef, big, and effective cones organise the geometry of higher-dimensional varieties. The Birkar-Cascini-Hacon-McKernan theorem (2010) — existence of minimal models / finite generation of canonical rings for varieties of general type — relies on systematic positivity arguments rooted in the theory developed in EGA II and Lazarsfeld's Positivity in Algebraic Geometry. Ampleness in arithmetic (height theory, Arakelov geometry, $p$ -adic positivity) and in derived algebraic geometry (Lurie) extends the framework to richer settings. Robert Lazarsfeld's two-volume Positivity in Algebraic Geometry (2004) is the modern canonical reference and remains under active extension.

Bibliography [Master]

Hartshorne, Algebraic Geometry — §II.7 (ample and very ample), §III.5 (cohomology of ample bundles).
Vakil, The Rising Sea: Foundations of Algebraic Geometry — §15–§16 (ample line bundles, projective embeddings).
Lazarsfeld, Positivity in Algebraic Geometry I — the modern canonical reference for ampleness, nef, big, and Seshadri constants.
Lazarsfeld, Positivity in Algebraic Geometry II — multiplier ideals, asymptotic invariants, advanced positivity.
Grothendieck-Dieudonné, Éléments de Géométrie Algébrique II (EGA II, 1961) — the foundational scheme-theoretic treatment of ampleness.
Serre, Faisceaux Algébriques Cohérents (FAC, 1955) — twisting sheaves and Serre's vanishing.
Kodaira, "On Kähler varieties of restricted type" (1954) — Kodaira embedding and vanishing.
Nakai, "Non-degenerate divisors on an algebraic surface" (1963) — numerical criterion (surface case).
Moishezon, "On $n$ -dimensional compact varieties with $n$ algebraically independent meromorphic functions" (1966) — projectivity criterion.
Kollár-Mori, Birational Geometry of Algebraic Varieties — ampleness in the minimal model program.
Eisenbud-Harris, 3264 and All That — ampleness and intersection theory.
Kleiman, "Toward a numerical theory of ampleness" (1966) — Kleiman's nef criterion.

Prerequisites

04.05.03
04.06.02
04.07.01

Used in

04.05.02

Tier anchors

beginner: Strogatz analogous level for line bundles inducing projective embeddings
intermediate: Hartshorne §II.7; Vakil §15–§16; Lazarsfeld *Positivity* Vol I
master: Hartshorne §II–§III; Vakil; Lazarsfeld *Positivity in Algebraic Geometry* Vol I; Grothendieck EGA II

References

TODO_REF
Hartshorne — Algebraic Geometry · §II.7, ample and very ample line bundles
TODO_REF
Vakil — The Rising Sea · §15–§16, ample line bundles, projective embeddings
TODO_REF
Serre — Faisceaux Algébriques Cohérents (FAC) · 1955, twisting sheaves and the Serre vanishing
TODO_REF
Grothendieck-Dieudonné — EGA II · 1961, ample and very ample formalism
TODO_REF
Lazarsfeld — Positivity in Algebraic Geometry · Vol I, ample line bundles and positivity criteria

Lean module

Codex.AlgebraicGeometry.Divisors.Ample

Reviewer

TBD

Estimated time

beginner: 12m
intermediate: 40m
master: 55m