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Ample line bundle

In algebraic geometry, an ample line bundle on a quasi-compact scheme is an invertible sheaf \mathcal{L} such that the associated sheaf of graded algebras \bigoplus_{d \geq 0} H^0(X, \mathcal{L}^{\otimes d}) is finitely generated, and the natural morphism X \to \Proj \bigoplus_{d \geq 0} H^0(X, \mathcal{L}^{\otimes d}) is an isomorphism onto its image, effectively embedding X projectively via sections of high tensor powers.^[1] Equivalently, \mathcal{L} is ample if there exist sufficiently many global sections in some power \mathcal{L}^{\otimes d} whose principal open sets X_s (for s \in H^0(X, \mathcal{L}^{\otimes d})) form an affine cover of X.^[1] This concept captures a notion of positivity central to the study of projective varieties and schemes, generalizing the role of the hyperplane bundle \mathcal{O}_{\mathbb{P}^n}(1) on projective space, which is ample and generates embeddings.^[2] Introduced in the mid-20th century, the term originates from Jean-Pierre Serre's work, where he characterized ample line bundles cohomologically: \mathcal{L} is ample if, for every coherent sheaf \mathcal{F} on X, there exists n_0 > 0 such that for all n \geq n_0, \mathcal{F} \otimes \mathcal{L}^{\otimes n} is generated by global sections.^[3] This criterion aligns with geometric ampleness, as high powers ensure vanishing of higher cohomology groups H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0 for i > 0 and n \gg 0, by Serre's vanishing theorem.^[2] Ample line bundles satisfy several equivalent criteria, including numerical ones like the Nakai-Moishezon condition (intersection numbers positive on subvarieties) over fields of characteristic zero, and metric conditions over the complex numbers, where \mathcal{L} admits a hermitian metric with positive curvature.^[2] They are preserved under tensor products with flexible sheaves (those generated by global sections) and remain ample when restricted to closed subschemes or pulled back under proper morphisms.^[1] On curves, ampleness reduces to positive degree, while on higher-dimensional varieties, it ensures the variety is projective.^[3] The notion extends to vector bundles, where a bundle E is ample if its projectivization carries an ample tautological line bundle, facilitating generalizations like ample subvarieties and q-ample divisors for partial cohomology vanishing.^[2] Ample bundles underpin key results in birational geometry, such as the basepoint-free theorem and Kodaira embedding theorem, and play a foundational role in moduli problems and positivity studies in algebraic geometry.^[3]

Introduction

Projective varieties and line bundles

A projective variety over an algebraically closed field k is defined as an algebraic variety that is isomorphic to a closed subvariety of some projective space \mathbb{P}^n_k.^[4] This construction embeds the variety into a space where homogeneous coordinates allow for a compact, well-behaved geometry, facilitating the study of global properties through polynomial equations.^[4] Projective varieties are integral, meaning they are irreducible and reduced, and they provide a foundational setting for much of classical algebraic geometry.^[5] Line bundles on a variety X are invertible sheaves \mathcal{L}, which are locally free \mathcal{O}_X-modules of rank one, equipped with transition functions that are invertible elements in the structure sheaf.^[6] The global sections H^0(X, \mathcal{L}) form a finite-dimensional vector space over k when X is projective, enabling the bundle to encode geometric data such as divisors.^[6] The Picard group \mathrm{Pic}(X) classifies isomorphism classes of line bundles under tensor product, and it is closely related to the divisor class group, where each line bundle corresponds to the class of a Cartier divisor via the map \mathcal{O}_X(D) \cong \mathcal{L}.^[7] On a smooth projective curve, the degree of a line bundle \mathcal{L} \cong \mathcal{O}_X(D) is defined as the degree of the divisor D, providing an invariant that measures the bundle's "size" and influences its section space.^[8] The study of line bundles on curves was motivated by early work in complex analysis and algebraic geometry, particularly the Riemann-Roch theorem, which relates the dimension of global sections of a line bundle to its degree and the genus of the curve, thereby guiding the construction of embeddings into projective space.^[9] This theorem, originally formulated by Riemann in 1857 and proved by Roch in 1865, highlighted how sufficiently positive line bundles could yield enough sections to embed the curve projectively.^[10] In general, for a projective variety X and a line bundle \mathcal{L} with a basis \{s_0, \dots, s_n\} of global sections, these sections induce a morphism \phi_{\mathcal{L}}: X \to \mathbb{P}^n_k defined by x \mapsto [s_0(x) : \dots : s_n(x)], where the map is well-defined away from the base locus of the sections.^[11] Such morphisms preserve the projective structure and allow line bundles to serve as tools for realizing varieties within projective space.^[11]

Very ample bundles and embeddings

A line bundle \mathcal{L} on a projective variety X over an algebraically closed field is very ample if the associated morphism \phi_{|\mathcal{L}|}: X \to \mathbb{P}^N, where N = h^0(X, \mathcal{L}) - 1, defined by the complete linear system |\mathcal{L}| is a closed embedding. This morphism sends a point x \in X to the line in H^0(X, \mathcal{L})^* consisting of sections vanishing at x, or equivalently, to the projective coordinates given by a basis of global sections evaluated at x. Equivalently, on a projective variety X, a line bundle \mathcal{L} is very ample if and only if there exists a closed embedding i: X \hookrightarrow \mathbb{P}^N such that \mathcal{L} \cong i^* \mathcal{O}_{\mathbb{P}^N}(1), the pullback of the tautological hyperplane bundle on projective space. This characterization underscores the role of very ample bundles as the strongest form of positivity, directly realizing X as a projective subvariety via its sections. A canonical example occurs on projective space \mathbb{P}^n, where the line bundle \mathcal{O}_{\mathbb{P}^n}(d) for any integer d \geq 1 is very ample. For d = 1, it induces the identity embedding \mathbb{P}^n \hookrightarrow \mathbb{P}^n. For d > 1, the morphism is the Veronese embedding v_d: \mathbb{P}^n \hookrightarrow \mathbb{P}^{\binom{n+d}{d}-1}, which maps a point [x_0 : \cdots : x_n] to the point whose coordinates are all monomials of degree d in the x_i. The morphism \phi_{|\mathcal{L}|} is defined using the complete linear system |\mathcal{L}|, the projectivization of the vector space of global sections H^0(X, \mathcal{L}), assuming |\mathcal{L}| is basepoint-free so that the map is defined everywhere on X. To see that \phi_{|\mathcal{L}|} is an embedding, it suffices to verify injectivity and that it is a closed immersion; the latter follows from projectivity. Injectivity arises because the global sections separate points and tangent vectors: for distinct points p, q \in X, there exists s \in H^0(X, \mathcal{L}) such that s(p) = 0 but s(q) \neq 0 (or vice versa), and for any p \in X and nonzero tangent vector v \in T_p X, there exists s \in H^0(X, \mathcal{L}) with s(p) = 0 but the differential ds_p(v) \neq 0.

Progression to ample bundles

While very ample line bundles on a projective variety provide a closed embedding into projective space, many line bundles that intuitively behave "positively" fail to be very ample themselves, as their global sections may not separate points or tangent vectors sufficiently. However, tensor powers of such bundles often rectify this limitation, achieving very ampleness for sufficiently high exponents and thereby embedding the variety projectively. This observation motivates a broader notion of positivity in algebraic geometry, shifting focus from immediate embeddability to asymptotic behavior under tensoring. A line bundle \mathcal{L} on a projective variety X is thus defined to be ample if there exists a positive integer k such that \mathcal{L}^{\otimes k} is very ample. This criterion, introduced by Serre, relaxes the stringent embedding condition of very ampleness (where k = 1 suffices) to one where higher powers embed X, capturing a wider class of line bundles that generate the Picard group and facilitate key vanishing results. Central to this progression is Serre's vanishing theorem, which states that if \mathcal{L} is ample on X, then for any coherent sheaf \mathcal{F} on X, the higher cohomology groups H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes k}) = 0 for all i > 0 and sufficiently large k. This cohomological property underscores the positivity of ample bundles, ensuring that high powers become globally generated and basepoint-free. Complementing this, the Nakai-Moishezon criterion provides a numerical perspective, roughly asserting that \mathcal{L} is ample if its intersection numbers with subvarieties are positive—specifically, (\mathcal{L}^{\dim V} \cdot V) > 0 for every subvariety V \subseteq X—highlighting geometric ampleness through intersection theory (detailed further in characterizations).

Definitions

Ample on projective varieties

In the classical setting, consider a projective variety X over an algebraically closed field k. A line bundle \mathcal{L} on X is called ample if there exists a positive integer n such that \mathcal{L}^{\otimes n} is very ample.^[12] This condition is equivalent to the existence of some n > 0 such that the morphism \phi_{|\mathcal{L}^{\otimes n}|}: X \to \mathbb{P}^N, where N = h^0(X, \mathcal{L}^{\otimes n}) - 1 and \phi_{|\mathcal{L}^{\otimes n}|} is induced by the complete linear system |\mathcal{L}^{\otimes n}|, is a closed embedding, with \mathcal{L}^{\otimes n} \cong \phi_{|\mathcal{L}^{\otimes n}|}^* \mathcal{O}_{\mathbb{P}^N}(1).^[12] Ample line bundles are closed under tensor products: if \mathcal{L} and \mathcal{M} are ample on X, then so is \mathcal{L} \otimes \mathcal{M}. To see this, choose positive integers r, s such that \mathcal{L}^{\otimes r} and \mathcal{M}^{\otimes s} are very ample; then (\mathcal{L} \otimes \mathcal{M})^{\otimes rs} = (\mathcal{L}^{\otimes r})^{\otimes s} \otimes (\mathcal{M}^{\otimes s})^{\otimes r}, and the tensor product of very ample line bundles is very ample since it corresponds to the pullback of \mathcal{O}_{\mathbb{P}^N}(a) \otimes \mathcal{O}_{\mathbb{P}^M}(b) \cong \mathcal{O}_{\mathbb{P}^N \times \mathbb{P}^M}(a, b), which is very ample relative to the product embedding.^[12] Moreover, ample line bundles are preserved under pullback along projective morphisms: if f: Y \to X is a morphism of projective varieties and \mathcal{L} is ample on X, then f^* \mathcal{L} is ample on Y. This follows from the fact that such pullbacks preserve the very ampleness of sufficiently high tensor powers, as projective morphisms between projective varieties admit factorizations involving projective bundles where the property holds by the definition.^[13] By definition, ample line bundles on varieties are invertible sheaves of rank one.^[12]

General definition on schemes

In the scheme-theoretic setting, the notion of an ample invertible sheaf extends the classical definition from projective varieties to more general quasi-compact and quasi-separated schemes. Let X be a quasi-compact quasi-separated scheme and \mathcal{L} an invertible sheaf on X. Following the approach in Grothendieck's Éléments de géométrie algébrique (EGA), \mathcal{L} is ample if there exists an integer n > 0 such that the natural morphism X \to \Proj_X(\bigoplus_{k \geq 0} \mathcal{L}^{\otimes kn}) is a closed immersion, where \Proj_X denotes the relative Proj construction over X applied to the associated graded sheaf of \mathcal{O}_X-algebras \bigoplus_{k \geq 0} \mathcal{L}^{\otimes k}.^[14] This condition ensures that X embeds as a closed subscheme into a projective bundle over itself, generalizing the embedding into projective space for varieties. Equivalently, the sets X_f = \{x \in X \mid f(x) \neq 0\} for homogeneous elements f \in \Gamma(X, \mathcal{L}^{\otimes m}) with m > 0 form an affine open cover of X.^[15] An equivalent formulation, emphasized in Hartshorne's Algebraic Geometry, defines \mathcal{L} as ample on a noetherian scheme X if, for every coherent sheaf \mathcal{F} on X, there exists an integer n_0 \geq 0 (depending on \mathcal{F}) such that \mathcal{F} \otimes \mathcal{L}^{\otimes n} is globally generated for all n \geq n_0.^[16] This cohomological criterion highlights ampleness as a positivity condition ensuring that tensor powers of \mathcal{L} "generate" the category of coherent sheaves asymptotically. For quasi-compact schemes, ampleness requires X to admit such an \mathcal{L}, and the two definitions (Proj embedding and global generation) coincide under mild assumptions like noetherianness or quasi-separatedness.^[17] For a proper scheme X over a field k via the structure morphism f: X \to \Spec k, an invertible sheaf \mathcal{L} on X is ample if the global sections H^0(X, \mathcal{L}^{\otimes n}) induce a closed immersion X \hookrightarrow \mathbb{P}^N_k for sufficiently large n, where N = \dim H^0(X, \mathcal{L}^{\otimes n}) - 1.^[17] This reduces to the classical case where ampleness implies X is projective over k, as the global sections H^0(X, \mathcal{L}^{\otimes n}) provide the embedding coordinates. In this setting, the Proj construction over \Spec k yields a closed immersion X \hookrightarrow \mathbb{P}^N_k for some N, up to tensor power. A key consequence is that if an ample invertible sheaf \mathcal{L} exists on X, then X is projective over its base (e.g., separated and proper over \Spec k), as the closed immersion into the Proj forces properness and separatedness.^[16] This implication holds more generally for morphisms, where relative ampleness ensures projectivity relative to the base.

Basic properties of ample bundles

Ample line bundles satisfy several fundamental algebraic properties that highlight their role in ensuring projectivity and positivity on schemes. If \mathcal{L} and \mathcal{M} are ample invertible sheaves on a scheme X, then their tensor product \mathcal{L} \otimes \mathcal{M} is also ample.^[3] Moreover, if \mathcal{L} is ample, then so is \mathcal{L}^{\otimes k} for every integer k \geq 1.^[18] Pullbacks preserve ampleness under certain morphisms. Specifically, for a finite surjective morphism f: Y \to X of proper schemes over a Noetherian ring, an invertible sheaf \mathcal{L} on X is ample if and only if its pullback f^*\mathcal{L} is ample on Y.^[19] Similarly, under an open immersion i: U \to X, if \mathcal{L} is ample on X, then the restriction i^*\mathcal{L} (or \mathcal{L}|_U) is ample on U.^[20] On smooth projective varieties over an algebraically closed field, there is a direct correspondence between ample line bundles and divisors via the degree on curves: an invertible sheaf \mathcal{L} is ample if and only if, for every irreducible curve C \subset X, the degree of \mathcal{L}|_C is positive.^[21] In the Picard group, the set of ample classes forms an open cone in the real Néron-Severi group N^1(X)_{\mathbb{R}}.^[22] Finally, on an affine scheme, no line bundle is ample.^[16]

Globally generated and basepoint-free sheaves

A sheaf \mathcal{F} of \mathcal{O}_X-modules on a scheme X is globally generated if the natural map \mathcal{O}_X \otimes_{\mathbb{Z}} H^0(X, \mathcal{F}) \to \mathcal{F} is surjective.^[23] This means that at every point x \in X, the evaluation map on stalks \mathcal{O}_{X,x} \otimes H^0(X, \mathcal{F}) \to \mathcal{F}_x is surjective, so the global sections generate \mathcal{F} locally as an \mathcal{O}_X-module.^[24] For a line bundle L on X, global generation is equivalent to the complete linear system |L| being basepoint-free, meaning that for every point x \in X, there exists a global section s \in H^0(X, L) that does not vanish at x.^[24] In other words, the sections have no common zeros, ensuring that the fibers of L are generated by global sections at every point.^[11] On a projective variety X over a field, a line bundle L is globally generated if and only if there exists a morphism \phi: X \to \mathbb{P}^N (for some N) such that L \cong \phi^* \mathcal{O}_{\mathbb{P}^N}(1).^[11] This pullback characterization highlights that global generation corresponds to L being the pullback of the tautological line bundle on some projective space, without requiring the morphism to be an embedding. Given a globally generated line bundle L on a projective variety X with r = \dim H^0(X, L) \geq 1, the global sections define a morphism \phi_L: X \to \mathbb{P}^{r-1} via the map sending x \in X to the line in H^0(X, L)^* consisting of sections vanishing at x.^[11] This morphism is well-defined everywhere precisely because L is basepoint-free, and it satisfies L \cong \phi_L^* \mathcal{O}_{\mathbb{P}^{r-1}}(1). A canonical example is the tautological line bundle \mathcal{O}_{\mathbb{P}^n}(1) on projective space \mathbb{P}^n, whose global sections are the linear forms x_0, \dots, x_n, which generate the stalk at every point since no hyperplane contains all of \mathbb{P}^n.^[11] Very ample line bundles are globally generated, as their sections embed the variety into projective space.^[11]

Nef and semi-ample line bundles

A nef line bundle provides a numerical criterion for positivity that weakens the geometric conditions of ampleness. On a projective variety X, a line bundle L is nef if its first Chern class satisfies c_1(L) \cdot C \geq 0 for every irreducible curve C \subset X. This condition captures the idea that L intersects curves non-negatively, reflecting a form of "non-negativity" in the intersection theory on X. The concept of a semi-ample line bundle introduces an asymptotic weakening, focusing on the behavior of powers of L. Specifically, L is semi-ample if there exists a positive integer k such that L^{\otimes k} is globally generated, meaning the natural evaluation map H^0(X, L^{\otimes k}) \otimes \mathcal{O}_X \to L^{\otimes k} is surjective. Equivalently, some power L^{\otimes k} is basepoint-free, ensuring that the complete linear system |L^{\otimes k}| defines a morphism from X to projective space without base points. Globally generated sheaves, such as these powers, are referenced as a prerequisite for this generation property. Asymptotically, L is semi-ample if and only if the morphism X \to \mathrm{Proj}\left( \bigoplus_{k \geq 0} H^0(X, L^{\otimes k}) \right) defined by the associated graded algebra is projective. On projective varieties, these notions form a chain of implications: every ample line bundle is semi-ample, and every semi-ample line bundle is nef.^[25] The first follows since powers of an ample bundle are very ample, hence globally generated; the second holds because a globally generated line bundle induces a morphism to projective space on which curve degrees are non-negative, implying nefness.^[25] These implications are strict in general, with equality holding in the Kähler case under the numerical equivalence for positivity.^[26] However, nef does not imply semi-ample; for instance, on the blowup of \mathbb{P}^2 at a point, certain boundary classes in the nef cone, such as limits of pullback ample bundles adjusted by the exceptional divisor, yield nef bundles whose powers remain non-globally generated.^[27]

Big line bundles

In algebraic geometry, a line bundle L on a projective variety X of dimension n is defined to be big if its volume is positive, where the volume is given by

\vol(L) = \lim_{k \to \infty} \frac{h^0(X, L^{\otimes k})}{k^n} > 0.

This limit exists and equals the leading coefficient of the Hilbert polynomial of L.^[28] The notion of bigness generalizes ampleness, as every ample line bundle is big, since the space of sections of high powers of an ample bundle grows polynomially with degree exactly n.^[28] On projective varieties, bigness admits a numerical characterization analogous to the Nakai-Moishezon criterion for ampleness: L is big if and only if for every irreducible subvariety Y \subseteq X of positive dimension p, the intersection number (c_1(L)^p \cdot Y) > 0.^[28] This condition captures the idea that L "fills up" the variety in a strong positivity sense across all dimensions. However, big line bundles need not be nef themselves. For example, on a projective surface, one can construct a big line bundle L with L^2 > 0 (ensuring bigness via the surface criterion) but L \cdot C < 0 for some irreducible curve C, such as in blow-up models where L has negative degree on an exceptional divisor while maintaining overall positive self-intersection.^[29] The cone of big classes in the Néron-Severi space, known as the big cone, forms the interior of the pseudoeffective cone, so its closure generates the pseudoeffective cone, which contains all effective divisor classes.^[30]

Criteria and Characterizations

Cohomological criteria

One key cohomological criterion for ampleness is provided by Serre's theorem, which characterizes ample line bundles on projective varieties in terms of vanishing of higher cohomology groups. Specifically, for a projective variety X over a field and a line bundle \mathcal{L} on X, \mathcal{L} is ample if and only if for every coherent sheaf \mathcal{F} on X and every i > 0, the cohomology group H^i(X, \mathcal{L}^{\otimes k} \otimes \mathcal{F}) = 0 for all sufficiently large k. The direction that ampleness implies such vanishing is known as Serre's vanishing theorem.^[31] In the complex analytic setting, Kodaira's vanishing theorem offers a related criterion, stating that if X is a compact complex manifold equipped with a Kähler metric and \mathcal{L} is an ample holomorphic line bundle on X, then H^i(X, \Omega_X^j \otimes \mathcal{L}) = 0 for all i + j > \dim X and i, j \geq 0, or equivalently in the case j = \dim X, H^i(X, K_X \otimes \mathcal{L}) = 0 for i > 0, where K_X is the canonical bundle. This theorem relies on the existence of a positive metric on \mathcal{L} and Hodge theory to establish the vanishing via Bochner-type arguments.^[32] A refinement in the complex case is the Akizuki-Nakano vanishing theorem, which strengthens Kodaira's result by asserting that if \mathcal{L} is an ample holomorphic line bundle on a compact complex manifold X, then the Dolbeault cohomology groups H^{p,q}(X, \mathcal{L}) = 0 for p + q > \dim X.^[33] For the general scheme-theoretic setting, the cohomological criterion extends to proper schemes over an affine base via pushforward considerations: a line bundle \mathcal{L} on a proper scheme f: X \to \operatorname{Spec}(A) with A affine is ample if and only if for every coherent sheaf \mathcal{F} on X and i > 0, H^i(X, \mathcal{L}^{\otimes k} \otimes \mathcal{F}) = 0 for k \gg 0, which implies that the higher direct images R^i f_*(\mathcal{L}^{\otimes k} \otimes \mathcal{F}) = 0 on \operatorname{Spec}(A).^[34] The proof of Serre's theorem leverages Castelnuovo-Mumford regularity, a notion that quantifies the minimal twist required for a coherent sheaf to have vanishing higher cohomology; for an ample line bundle, high powers ensure regularity for any coherent sheaf, leading to the desired vanishing.^[35]

Numerical and intersection criteria

Numerical criteria for the ampleness of a line bundle L on a projective variety X rely on intersection theory, where the first Chern class c_1(L) is considered in the Chow group A^*(X) or the Néron-Severi group, and intersection products are defined via the ring structure on the Chow ring, associating to cycles their degrees when the dimension matches that of X. These products allow numerical invariants like self-intersections and mixed intersections to test positivity properties essential for ampleness. The Nakai-Moishezon criterion provides a comprehensive intersection-theoretic characterization: a line bundle L on a projective variety X of dimension n is ample if and only if for every irreducible subvariety Y \subseteq X, the intersection number (c_1(L)^{\dim Y} \cdot Y) > 0. This condition ensures that L restricts positively on all subvarieties, capturing the global positivity required for ampleness. Originally established for surfaces by Nakai and extended to higher dimensions by Moishezon, the criterion generalizes the basic case on curves, where L is ample if and only if its degree \deg(L) = c_1(L) \cdot [C] > 0 for the curve C = X. Kleiman's criterion offers an equivalent formulation using mixed intersections with a fixed ample bundle: L is ample if and only if, for some (equivalently, every) ample line bundle H on X, the intersection numbers c_1(L)^i \cdot c_1(H)^{n-i} > 0 for all i = 1, \dots, n.^[36] In particular, this implies the top self-intersection c_1(L)^n > 0. This criterion highlights the position of c_1(L) in the interior of the ample cone in the Néron-Severi space, providing a practical test via comparisons with known ample classes.^[36] These criteria extend to projective schemes, including singular varieties, through the use of cycle classes in the Chow groups, with refinements in the 1980s confirming their validity without requiring smoothness assumptions, as developed in the framework of intersection theory on singular spaces.

Openness and stability properties

The set of ample classes in the Néron–Severi real vector space N^1(X)_{\mathbb{R}} of a projective variety X forms an open convex cone, known as the ample cone \operatorname{Amp}(X). This openness follows from Kleiman's criterion, which characterizes ampleness numerically via positive intersections with all subvarieties, combined with the closedness of the dual Kleiman–Mori cone of effective curves. The proof relies on the semicontinuity of intersection forms under flat morphisms. Specifically, for a fixed effective cycle class \alpha \in N_k(X), the pairing \langle c_1(L), \alpha \rangle is upper semicontinuous in families of line bundles L, ensuring that positivity conditions defining ampleness hold in an open neighborhood. This semicontinuity arises from the behavior of Chern classes and cycle classes in flat families, preserving the strict inequality for ample classes nearby. (Hartshorne's Algebraic Geometry discusses related semicontinuity for cohomology, underpinning intersection theory.) Ampleness is stable under small deformations of the variety. If L is ample on X, then for a flat family \mathcal{X} \to B with X = \mathcal{X}_b and a line bundle \mathcal{L} on \mathcal{X} restricting to L on X, \mathcal{L} remains ample on fibers \mathcal{X}_{b'} for b' sufficiently close to b. This preservation stems from the openness of the ample cone and the upper semicontinuity of the Néron–Severi rank, ensuring the deformed class stays in the interior.^[37] Kleiman further characterized ampleness asymptotically: a line bundle L on X is ample if and only if its asymptotic intersection numbers with every effective cycle are positive, meaning \lim_{m \to \infty} \frac{1}{m^k} (c_1(L)^k \cdot Z) > 0 for every k-dimensional subvariety Z. Post-2000 research has extended these properties to ample cones in moduli spaces of vector bundles. For instance, in the moduli space of Gieseker semistable sheaves on \mathbb{P}^2, the ample cone is generated by specific tautological bundles and admits a chamber structure determined by Bridgeland stability conditions, reflecting walls where stability flips occur.^[38] Similarly, on K3 surfaces, the ample cone of the moduli space of stable sheaves decomposes into chambers corresponding to different polarization types, with ampleness preserved across Noether–Lefschetz loci under deformations.^[39]

Other characterizations

A line bundle \mathcal{L} on a projective scheme X over an algebraically closed field k is ample if and only if the natural morphism X \to \Proj\bigoplus_{k \geq 0} \mathcal{L}^{\otimes k} induced by the surjection from the symmetric algebra \Sym \mathcal{L}^\vee to the graded algebra \bigoplus_{k \geq 0} H^0(X, \mathcal{L}^{\otimes k}) is a closed immersion after passing to a sufficiently high Veronese subring, i.e., for some m \gg 0, the morphism to \Proj\bigoplus_{d \geq 0} H^0(X, \mathcal{L}^{\otimes (md)}) is a closed immersion. This characterization extends the Proj construction used for very ample bundles, where the full graded algebra suffices without Veronese regrading. Equivalently, \mathcal{L} is ample if and only if there exists an integer m \gg 0 such that m\mathcal{L} is very ample, i.e., the morphism f: X \to \mathbb{P}^N_k associated to the complete linear system |m\mathcal{L}| is a closed immersion with m\mathcal{L} \cong f^* \mathcal{O}_{\mathbb{P}^N_k}(1), ensuring that \mathcal{L} inherits positivity from the tautological ample bundle \mathcal{O}(1) via tensor powers. In this setup, the morphism f embeds X projectively while preserving ampleness under pullback. Another characterization arises from the geometry of sections: \mathcal{L} is ample if and only if the complete linear system |\mathcal{L}^{\otimes k}| separates points and tangent vectors on X for all sufficiently large k, meaning that for distinct points p, q \in X, there exists a section s \in H^0(X, \mathcal{L}^{\otimes k}) vanishing at p but not at q, and for any p \in X and tangent vector v \in T_p X, there exists s vanishing at p whose differential does not annihilate v. This separation property ensures that high powers of \mathcal{L} define embeddings into projective space, mirroring the defining feature of very ampleness but relaxed to asymptotic behavior. Abstractly, in the Néron-Severi space N^1(X)_{\mathbb{R}}, the class [\mathcal{L}] generates the ample cone \Amp(X) positively in the sense that the ample cone consists of all finite positive \mathbb{R}-linear combinations of classes of ample line bundles like [\mathcal{L}], forming an open convex cone whose interior captures ampleness. This numerical perspective aligns the geometric notion with intersection theory, where ampleness corresponds to positive generation within the cone dual to the Mori cone of curves. These equivalences hold over algebraically closed fields, where cohomological and numerical criteria align seamlessly; over general bases, subtleties arise, such as the need for base change to verify separation or immersion properties faithfully.

Examples

Positive examples from classical geometry

In classical algebraic geometry, the projective space \mathbb{P}^n over an algebraically closed field provides a fundamental example of ample line bundles. The tautological line bundle \mathcal{O}_{\mathbb{P}^n}(1) is very ample, as it realizes the identity embedding of \mathbb{P}^n into itself via the complete linear system of its global sections, which are the homogeneous linear coordinates.^[40] Consequently, \mathcal{O}_{\mathbb{P}^n}(d) is ample for every positive integer d > 0, since powers of very ample (hence ample) line bundles remain ample.^[40] On elliptic curves, which are smooth projective curves of genus one, every line bundle of positive degree exemplifies ampleness. For a complete nonsingular curve C over a field k, a line bundle is ample if and only if its degree is positive; this holds in particular for elliptic curves, where the degree condition ensures that high tensor powers generate global sections sufficient to embed the curve projectively.^[21] Thus, any line bundle L on an elliptic curve E with \deg L > 0 is ample.^[21] Abelian varieties offer translation-invariant ample line bundles as key examples. On an abelian variety A over a field k, a line bundle L is ample if it is nondegenerate, meaning the kernel K(L) of the induced map on the dual abelian variety is finite.^[41] A prominent case is the principal polarization, where L has a unique effective divisor up to translation, known as the theta divisor \Theta; the associated line bundle \mathcal{O}_A(\Theta) is ample and translation-invariant, inducing an isogeny \phi_L: A \to \hat{A} of degree equal to the dimension of its space of global sections.^[42] The Grassmannian \mathrm{Gr}(k, n), parametrizing k-dimensional subspaces of an n-dimensional vector space V, features the Plücker line bundle as a canonical ample example. This bundle, denoted \mathcal{O}_{\mathrm{Gr}(k,n)}(1) or the determinant of the tautological quotient bundle, generates the Picard group \mathbb{Z} and is very ample, embedding \mathrm{Gr}(k,n) into \mathbb{P}(\wedge^k V) via Plücker coordinates that map each k-plane to the projectivized wedge product of a basis.^[43] Toric varieties illustrate ample line bundles through sums of torus-invariant divisors. On a projective toric variety X_\Sigma defined by a fan \Sigma in \mathbb{Z}^d, a torus-invariant Cartier divisor D = \sum_{\rho \in \Sigma(1)} a_\rho D_\rho (with a_\rho > 0 for all rays \rho) corresponds to an ample line bundle \mathcal{O}_X(D) if its associated support function \phi_D is strictly convex on \Sigma, ensuring that the polyhedron P_D = \{ m \in M_\mathbb{R} \mid \langle m, u_\rho \rangle \geq -a_\rho \ \forall \rho \} lies in the strictly positive orthant and intersects every maximal cone.^[44] For instance, on \mathbb{P}^n as a toric variety, the hyperplane divisor yields such an ample bundle.^[44]

Non-examples and boundary cases

A classic example of a nef line bundle that is not ample arises on the product of two projective lines, \mathbb{P}^1 \times \mathbb{P}^1. The line bundle \mathcal{O}(1,0) is the pullback of \mathcal{O}_{\mathbb{P}^1}(1) from the first factor via the projection. It is nef because its degree is non-negative on every curve, but it is not ample since its degree is zero on fibers of the second ruling, i.e., curves of bidegree (0,1).^[45] On the blowup X = \mathrm{Bl}_p \mathbb{P}^2 of \mathbb{P}^2 at a point p, the line bundle \mathcal{O}_X(H - E), where H is the pullback of \mathcal{O}_{\mathbb{P}^2}(1) and E \cong \mathbb{P}^1 is the exceptional divisor, provides another boundary case. This bundle is nef, as its first Chern class intersects every irreducible curve non-negatively: it has degree $1onE

, degree &#36;0

on the strict transforms of lines through p, and degree $1on the strict transforms of lines not passing throughp. However, it is not ample because its self-intersection is zero, so powers do not embed X$ projectively.^[46] K3 surfaces offer examples of effective divisors that are nef but fail to be ample. Consider a K3 surface S admitting an elliptic fibration over \mathbb{P}^1 with generic fiber F. The class [F] is effective (as the fibration is a union of effective fibers) and nef, since S has trivial canonical bundle and [F] has non-negative intersection with every curve. Yet [F] is not ample because its self-intersection vanishes: F^2 = 0. Such boundary cases occur precisely when the Picard rank of S is at least $2$, allowing for divisors orthogonal to an ample class in the Néron-Severi group.^[47] On non-projective varieties, such as the affine line \mathbb{A}^1, the trivial line bundle \mathcal{O}_{\mathbb{A}^1} illustrates a fundamental boundary. While \mathcal{O}_{\mathbb{A}^1} is trivial (hence locally ample on affines), it cannot be ample globally because ampleness requires some power to be very ample, embedding the scheme projectively into \mathbb{P}^N, but \mathbb{A}^1 admits no such projective embedding as it is affine and quasi-affine.^[16] For singular schemes, the distinction between nef and ample can be more pronounced than on smooth varieties. An example from the study of Hilbert schemes of points on a smooth surface shows that an ample line bundle on the singular symmetric product \mathrm{Sym}^n(S) (for n \geq 2) pulls back via the Hilbert-Chow morphism to a big and nef but not ample line bundle on the smooth Hilbert scheme S^{}, illustrating how singularities influence ampleness criteria.^[48]

Generalizations

Ample vector bundles

The notion of an ample vector bundle on a projective variety generalizes the concept of an ample line bundle to higher-rank settings, capturing positivity conditions that ensure generation of sections in high symmetric powers. This concept was introduced by Robin Hartshorne in 1966.^[3] For a vector bundle E of rank r \geq 1 on a scheme X, E is defined to be ample if the tautological line bundle \mathcal{O}_{P(E)}(1) on the projectivization P(E) is ample.^[3] This definition extends the case of line bundles, where r=1 and P(E) \cong X, so that ampleness of E coincides directly with that of the line bundle itself.^[3] A key characterization of ampleness relies on symmetric powers: E is ample if and only if, for every coherent sheaf \mathcal{F} on X, there exists an integer n_0 > 0 such that for all n \geq n_0, the sheaf \mathcal{F} \otimes \mathrm{Sym}^n(E) is globally generated.^[3] Equivalently, the symmetric powers \mathrm{Sym}^n(E) themselves are ample for sufficiently large n, and conversely, if \mathrm{Sym}^k(E) is ample for some k > 0, then E is ample.^[49] These properties ensure that ample vector bundles behave analogously to ample line bundles in terms of embedding varieties via sections. Ampleness is preserved under tensor products with ample line bundles: if E is an ample vector bundle and L is an ample line bundle on X, then E \otimes L is also ample.^[3] More generally, in characteristic zero, the tensor product of two ample vector bundles is ample.^[3] In the complex analytic setting on a projective manifold, if a holomorphic vector bundle admits a Hermitian metric whose curvature is positive in the sense of Griffiths (meaning the curvature tensor satisfies \Theta_h(u, \bar{u}) > 0 for all nonzero holomorphic vectors u), then the bundle is ample.^[50] The converse—that every ample vector bundle admits such a positively curved metric—remains a conjecture due to Griffiths.^[51] A classical example is the tangent bundle T\mathbb{P}^n on projective space \mathbb{P}^n, which is ample as it decomposes as a direct sum of ample line bundles (the relative \mathcal{O}(1) bundles over Grassmannians in its presentation).^[49] This ampleness reflects the high positivity of \mathbb{P}^n among projective varieties.

Q-ample and rational variants

In algebraic geometry, a Q-divisor on a projective variety X is defined to be ample if there exists a positive integer m such that mD is linearly equivalent to an ample Cartier divisor on X. This rational variant extends the classical notion of ampleness from integer coefficients to rational ones, allowing for finer control in numerical and intersection-theoretic criteria while preserving key positivity properties, such as the vanishing of higher cohomology groups for twists by sufficiently large multiples. For line bundles, the corresponding Q-ample notion applies to Q-Cartier divisors, where a line bundle L = \mathcal{O}_X(D) associated to a Q-Cartier divisor D is Q-ample if some tensor power L^{\otimes m} corresponds to an ample Cartier divisor.^[52] More precisely, D is Q-ample if mD is Cartier and ample for some m > 0, enabling the bundle to capture rational positivity that integer line bundles may not.^[53] This framework ensures that Q-ample line bundles behave analogously to ample ones in embedding theorems and generation of global sections, but over the rational Néron-Severi group. Ample Q-divisors and Q-ample line bundles generate the ample cone rationally: the interior of the ample cone in N^1(X)_\mathbb{Q} consists precisely of numerical classes of ample Q-divisors, providing a dense rational basis for the full real ample cone.^[25] For instance, on orbifold varieties, Q-ample divisors arise naturally in crepant resolutions, where the pullback of an ample Q-divisor from the orbifold to the resolved space remains Q-ample, facilitating curve counting invariants and equivalence between orbifold and resolved geometries.^[54] In the minimal model program for klt pairs, ample Q-divisors play a central role in scaling techniques and birational transformations.

Ampleness in derived categories

In the bounded derived category D^b(\coh X) of coherent sheaves on a projective scheme X, an object E is called ample if there exists an ample line bundle F on X such that for every object G \in D^b(\coh X), the higher cohomology groups of the complex \RHom(E, F^k \otimes G) vanish, i.e., H^i(\RHom(E, F^k \otimes G)) = 0 for all i \neq 0 and all sufficiently large k \gg 0.^[55] This condition generalizes the classical Serre vanishing theorem, where tensoring with high powers of an ample line bundle resolves higher cohomology, ensuring that E acts as a "projective-like" generator in high twists. The vanishing implies that \RHom(E, F^k \otimes -) is represented by its zeroth cohomology, making the functor exact and detecting the structure of the category through finite-dimensional Hom-spaces. Classical ample line bundles extend naturally to this setting: if L is an ample line bundle viewed as an object in D^b(\coh X), then taking F = L satisfies the condition, as \RHom(L, L^k \otimes G) \simeq \RHom(\mathcal{O}_X, L^{k-1} \otimes G), and Serre vanishing ensures the higher Ext groups \Ext^i(\mathcal{O}_X, L^{k-1} \otimes G) = 0 for i > 0 and k \gg 0.^[56] More generally, any shift L or direct summand of powers of ample line bundles inherits ampleness, bridging the abelian category of coherent sheaves to its derived enhancement. Ample objects play a key role in Fourier-Mukai transforms, where ampleness is preserved under such equivalences. For instance, on an abelian variety A, the symmetric Fourier-Mukai transform with Poincaré kernel maps an ample line bundle L on A to an ample vector bundle \hat{L} on the dual \hat{A}, with rank equal to h^0(A, L), maintaining the generating properties across the equivalence D^b(\coh A) \simeq D^b(\coh \hat{A})^\op.^[57] A fundamental theorem states that on a projective scheme X, ample objects generate the derived category: specifically, for an ample line bundle L, the thick triangulated subcategory generated by \{ L^{\otimes n} \mid n \geq 0 \} (under shifts, cones, and direct summands) is the entire D^b(\coh X), as the twists detect all nonzero objects via nonzero Hom-spaces in some degree.^[55] This generation occurs in at most \dim X + 1 steps for smooth projective varieties, extending Beilinson's resolution on projective space. Post-2010 developments incorporate the Balmer spectrum to study ampleness in tensor-triangulated categories. For a smooth projective variety X with ample canonical bundle \omega_X, the tensor structure \otimes_{L_X} on D^b(\coh X) induced by the line bundle tensor product is the unique such structure making \mathcal{O}_X the unit and \omega_X invertible, as determined by the Balmer spectrum \Spc(D^b(\coh X)) \cong X.^[58] This uniqueness relies on the spectrum classifying tensor ideals corresponding to supports, with ample objects ensuring the category is compactly generated and the spectrum recovers the underlying geometry.

Relative Ampleness

Definition in morphisms of schemes

In the setting of a morphism of schemes f: X \to S, where f is proper, a line bundle \mathcal{L} on X is defined to be f-ample (or relatively ample over S) if the restriction \mathcal{L}|_{X_s} is ample on every geometric fiber X_s = f^{-1}(s) for s \in S.^[59]^[60] This condition ensures that ampleness behaves fiberwise, leveraging the properness of f to guarantee that each fiber is a proper scheme over the residue field at s, on which the classical notion of ampleness applies.^[15] More generally, this is tied to the associated graded \mathcal{O}_S-algebra \mathcal{A} = \bigoplus_{n \geq 0} f_*(\mathcal{L}^{\otimes n}) being such that the canonical map X \to \mathrm{Proj}_f(\mathcal{A}) is a closed immersion over S.^[61] A standard example arises in the case of a projective bundle \pi: \mathbb{P}(E) \to S, where E is a locally free sheaf of finite rank on S; here, the tautological relative line bundle \mathcal{O}_{\mathbb{P}(E)}(1) is \pi-ample. This illustrates how relative ampleness extends the embedding properties of projective spaces to families over a base scheme S.^[60]

Criteria for relative ampleness

A line bundle \mathcal{L} on a scheme X is relatively ample with respect to a proper morphism f: X \to S if some power \mathcal{L}^{\otimes k} induces a closed immersion into a projective space bundle over S. This notion generalizes absolute ampleness, where S is a point, and analogous criteria exist for testing relative ampleness. One key cohomological test for relative ampleness draws from the absolute case, where Serre's criterion states that a line bundle is ample if higher cohomology groups vanish for sufficiently large powers twisted by arbitrary coherent sheaves. In the relative setting, a necessary condition is the fiberwise version: for every s \in S, the restriction \mathcal{L}|_{X_s} is ample on the fiber X_s, meaning H^i(X_s, \mathcal{L}^k|_{X_s}) = 0 for all i > 0 and k \gg 0. This fiberwise ampleness holds if and only if the higher direct images R^i f_*(\mathcal{L}^{\otimes k}) = 0 for i > 0 and k \gg 0, by the relative Serre vanishing theorem, which applies under suitable hypotheses such as f being projective.^[63]^[64] For a projective morphism f: X \to S, a line bundle \mathcal{L} on X is f-ample if and only if for every coherent sheaf \mathcal{F} on X, there exists m_0 > 0 such that R^i f_* (\mathcal{F} \otimes \mathcal{L}^{\otimes m}) = 0 for all i > 0 and m \geq m_0, and the natural evaluation map f^* f_* (\mathcal{F} \otimes \mathcal{L}^{\otimes m}) \to \mathcal{F} \otimes \mathcal{L}^{\otimes m} is surjective for m \gg 0. This ensures that the associated relative Proj construction yields an isomorphism over S, embedding X projectively relative to S.^[65] An intersection-theoretic test, analogous to Kleiman's numerical criterion in the absolute case, provides another characterization: \mathcal{L} is f-ample if its degree is positive on every nonzero effective relative cycle class in the Chow group of X/S. This relative Kleiman criterion holds for projective morphisms and implies that ampleness can be detected numerically via intersections with one-dimensional relative cycles.^[64]^[66] The set of relatively ample line bundles forms an open subset of the relative Picard scheme \operatorname{Pic}_{X/S}, provided it exists; this openness follows from the corresponding property for fiberwise ampleness and the continuity of pushforward operations under base change. Over a base S where the relative Picard functor is representable, the ample locus is the complement of the boundary of the relative ample cone in the Néron-Severi space.^[63]^[67] In the Hilbert scheme \operatorname{Hilb}^d_{P}(X/S) parameterizing flat families of subschemes of degree d and Hilbert polynomial P relative to S, with X projective over S via an ample \mathcal{O}_X(1), the determinant of the universal quotient bundle on the universal family is relatively ample over the Hilbert scheme. This line bundle \mathcal{O}_{\operatorname{Hilb}}(1) ensures the scheme is projective over S and plays a key role in embedding families of subschemes.^[68]^[69]

Applications to families and moduli

In the context of projective families, relative ampleness plays a crucial role in constructing and parameterizing subschemes via the Hilbert scheme. Consider a projective morphism X \to S equipped with a relatively ample line bundle \mathcal{O}_X(1). The Hilbert functor \mathrm{Hilb}_P(X/S), which parameterizes flat families of subschemes of X over S-schemes with Hilbert polynomial P, is representable by a projective scheme \mathrm{Hilb}_P(X/S) over S. This projectivity arises because the relative ampleness of \mathcal{O}_X(1) allows for a linearization of the action in the Grassmannian of quotients, bounding cohomology and enabling the use of flattening stratifications to ensure properness and finiteness.^[70] For moduli spaces of vector bundles, the ampleness of a line bundle on the moduli space is intimately tied to the stability condition of the bundles it parameterizes. Given a projective variety X and an ample line bundle H on X, the moduli space M_H(r, c_1, c_2) of H-stable vector bundles of rank r and Chern classes c_1, c_2 is a projective scheme, as established by the boundedness and openness of stability, which allow GIT quotients to yield projective varieties. Here, the ampleness of the determinant line bundle (or a suitable polarization) on M_H ensures the space is projective and reflects the μ-stability with respect to H, implying that points in the moduli correspond to bundles where no subsheaf destabilizes the slope. This connection is foundational, as varying H alters the stability notion and thus the moduli components. Ampleness is preserved under versal deformations in the setting of polarized varieties. For a polarized variety (X, L) with L ample, the versal deformation space carries a universal family \mathcal{X} \to T with a relative line bundle \mathcal{L} that restricts to L on the central fiber and remains relatively ample over T. This preservation follows from the openness of ampleness in flat families and the construction of the versal space, ensuring that nearby fibers inherit the projective embedding properties. In particular, for primitively polarized K3 surfaces, smooth versal L-deformations maintain the relative degree and ampleness of \mathcal{L}.^[71] A key example arises in Brill-Noether theory on the universal curve. Over the moduli space \mathcal{M}_g of genus g curves, the universal curve \mathcal{C} \to \mathcal{M}_g admits a relative ample line bundle \omega_{\mathcal{C}/\mathcal{M}_g}^\vee, the dualizing sheaf, which parameterizes linear series via the relative Picard scheme. Relative ampleness ensures the determinantal construction of Brill-Noether loci W^r_d \subset \mathrm{Pic}^d(\mathcal{C}) is proper and projective, allowing the expected dimension formula \rho(g,r,d) = g - (r+1)(g - d + r) to govern the geometry, with non-emptiness for general curves when \rho \geq 0. This setup linearizes the study of maps to projective space in families.^[72] Recent developments as of 2025 highlight the role of relative ampleness in mirror symmetry, particularly relating to Lagrangian fibrations. In relative mirror symmetry for log Calabi-Yau varieties (X, D) with anticanonical divisor D, a Lagrangian torus fibration on the complement X \setminus D mirrors a relative ample line bundle on the B-side, encoding wall-crossing phenomena and SYZ duality. This connection extends classical SYZ predictions, where the relative polarization corresponds to the monodromy of the fibration, facilitating homological mirror symmetry for non-Fano pairs.^[73]

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[PDF] Curve counting invariants for crepant resolutions - UBC Math
Dec 18, 2013 · an orbifold and its crepant resolution to be equivalent in some way. ... Let A be an ample Q-divisor with A · β = eH · β and let. A = (1 ...
[52]
Minimal model program - Project Euclid
Let (X, ∆) be a klt pair where KX + ∆ is pseudo-effective over S, ∆ = A+B, A is an ample effective Q-divisor, and B is an effective R-divisor. Then we have the ...
[53]
On the 4-dimensional minimal model program for Kähler varieties
In this paper we take the first steps towards proving that the minimal model program holds for Kähler 4-folds.
[54]
Section 36.16 (0BQQ): An example generator—The Stacks project
In this section we prove that the derived category of projective space over a ring is generated by a vector bundle, in fact a direct sum of shifts of the ...
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https://stacks.math.columbia.edu/tag/0BQQ
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[PDF] The Fourier-Mukai transform made easy
for ample line bundles, the symmetric Fourier-Mukai transform therefore in- terchanges “rank” and “dimension of the space of global sections”. The gen ...
[57]
Tensor triangulated category structures in the derived ... - MSP
Bondal and Orlov [2001] showed that if X is a smooth projective variety over C with ample ... Keywords: tensor triangulated category, Balmer spectrum, Bondal– ...
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Definition 29.37.1 (01VH)—The Stacks project
This leaves the possibility open to have a later definition of a relatively ample sheaf even in cases where the morphism is not quasi-compact. However, as far ...Missing: 4 | Show results with:4
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[PDF] Positivity in Algebraic Geometry I - ReadingSample - NET
From the contemporary viewpoint these so-called nef divisors lie at the heart of the theory of positivity for line bundles. ... Then Nef(X) = NE(X)∗ by definition ...
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Lemma 29.37.4 (01VJ)—The Stacks project
Assume (3) holds for the affine open covering S = \bigcup V_ i. We are going to show (5) holds. Since each f^{-1}(V_ i) has an ample invertible sheaf we ...
[61]
[PDF] relative ampleness in rigid geometry - Mathematics
The aim of this paper is to develop a rigid-analytic theory of relative ampleness for line ... ample line bundle and Theorem 3.2.7 to reduce to the case when X is ...
[62]
Toward a Numerical Theory of Ampleness - jstor
Toward a numerical theory of ampleness. By STEVEN L. KLEIMAN. TABLE OF ... KLEIMAN, A Note on the Nakai-Moisezon test for ampleness of a divisor, Amer ...
[63]
(Relative) ampleness on algebraic spaces - MathOverflow
Sep 19, 2013 · The notion of a relative ample (or f-ample) line bundle can be defined in several equivalent ways; Lazarsfeld's book gives the clearest discussion I have found.
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[PDF] The Picard scheme
Section 3 treats relative effective (Cartier) divisors on X/S and the relation of linear equivalence. We prove these divisors are parameterized by an open ...
[65]
[PDF] Math 249B. Hilbert schemes and a “universal” stable marked curve
Hilbert schemes classify flat families of closed subschemes, determined by a quasi-coherent ideal sheaf, and are related to the classification of closed ...
[66]
[PDF] Construction of Hilbert and Quot Schemes - arXiv
relatively very ample line bundle. ... is represented by the S-scheme P(E) = ProjSymOS E, with the tautological quotient π∗(E) → OP(E)(1) as the universal family.
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[PDF] the hilbert scheme
Let X → S be a projective scheme, O(1) a relatively ample line bundle and P a fixed polynomial. Recall that the Hilbert functor. HilbP (X/S) : {Schemes/S}o → { ...
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[PDF] arXiv:2109.12187v4 [math.AG] 13 May 2025
May 13, 2025 · We consider a versal F-deformation space of a primitively polarized K3 surface (X;L, R) with intersection lattice L2 = 2g − 2,R2 = −2,L · R = 2.
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[PDF] Brill-Noether duality for moduli spaces of sheaves on K3 surfaces
))(1) the relative ample line-bundle and let η : β∗ต ker(e|Mµ ) → β∗ต ... Denote by C ⊂ S×|L| the universal curve. The compactified relative Picard ...
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[PDF] Relative mirror symmetry for non-Fano varieties - arXiv
Oct 16, 2025 · Abstract. Given a smooth projective variety X with a smooth anticanonical divisor D, we study mirror symmetry for the log.