Picard and Jacobian varieties of algebraic curves, including relative constructions, Albanese maps, autoduality and Weil pairings. Includes Kummer theory and torsion constructions.
The group law and Abel–Jacobi map constructed from the represented Picard functor identify finite sums of geometric-fibre points with the classes of their normalized point divisors. Hence principal differences of those d…
For a pointed smooth proper geometrically integral curve model over an algebraically closed field, choose the actual group law and Abel–Jacobi morphism supplied by the Picard representation. Two morphisms from that repre…
For a pointed smooth proper geometrically integral curve over an arbitrary field, the actual Picard representation supplies an Abel–Jacobi map that detects morphisms to separated group schemes. Multiplication preservatio…
The actual Picard representation supplies an Abel–Jacobi map for which a pointed map into an abelian variety has at most one factorization among all scheme morphisms. Each candidate preserves the identity because its res…
For one represented Picard functor, two morphisms classifying the line bundle of each geometric point minus the basepoint agree on geometric points. The smooth source is reduced, so geometric point density proves equalit…
Riemann--Roch supplies a finite open cover of the Picard scheme by families of effective divisors of a common degree. Their sum morphisms agree on the actual overlaps, since their twists are restrictions of the same Poin…
Adding copies of the fixed base point makes a single moving point into a divisor of the degree used in the chart construction. Its normalized line bundle is the point-minus-basepoint bundle and its sum is the original po…
The factorization over a finite separable extension is identified with the native scalar extension of one fixed Abel--Jacobi morphism. Faithfully flat descent then gives a factorization over the original field.
The factorization over a finite separable extension descends along its finite faithfully flat spectrum map. Equality on the overlap follows from actual geometric point classes and uniqueness among all scheme morphisms.
+# The concrete divisor chart as an open subscheme of the Picard scheme The chart map classifies the actual twist of the universal divisor plus an auxiliary divisor. Its inverse onto an open subset of the representing Pi…
+# Sums on divisor Picard charts Over a field, equal-degree divisor twists are isomorphic exactly when their divisor line bundles are isomorphic: rigidification contributes only a trivial line from the base and the commo…
At a field-valued test map, rigidification changes a line bundle only by a trivial line from that test field. Tensor cancellation therefore identifies the divisor line bundles from an isomorphism of their actual twists.…
A morphism from a scheme to a relative group defines a sum morphism on every finite power of the source. Its action on test schemes is natural. Over an arbitrary field, properness and geometric integrality of the target…
The Picard construction supplies one designation, one representing Poincaré family, and one Abel--Jacobi morphism. The point-difference theorem and that same family's point comparison make the actual morphism bijective o…
The actual structure-sheaf cohomology comparison constructs a genus-one curve model. Its actual Abel--Jacobi isomorphism supplies a proper smooth connected commutative group. Inverting this same isomorphism constructs ev…
For an actual curve model and a chosen point, the point-difference map to Pic⁰ is bijective. Surjectivity constructs a point from a degree-one representative using Riemann--Roch; injectivity uses uniqueness of that point…
A closed point supplies a rational basepoint internally. The actual coherent genus comparison and Abel--Jacobi isomorphism then identify the original curve with its constructed Albanese variety. Pullback of finite etale…
The actual effective divisor x + n ε over any field-valued test scheme has normalized line bundle O(x - ε). Its actual divisor sum evaluates to q(x) when q sends the basepoint to the identity.
The curve, target group, universal divisor, and sum morphism are base changed along the same actual field-valued test map. The actual divisor equivalence and its inverse-ideal-module comparison transport the line bundles…
+# Finite sums depend only on the line bundle of an effective divisor The universal divisor scheme is fixed throughout. Canonical divisor sections, transported to a common invertible module, either define an independent…
+# Finite sums are constant along an actual linear pencil The independent pencil supplies a relative divisor over the affine line. Its actual classifying morphism into the universal divisor scheme, followed by the descen…
Adjoining the rational basepoint defines an actual morphism between successive finite fibre powers. A morphism sending that basepoint to the identity has the same finite sum before and after this insertion. This works ov…
Over any field, finite sums of the images of a smooth geometrically connected pointed curve in a proper geometrically integral group scheme descend through the actual universal effective-divisor map. Its finite flat kern…
Riemann--Roch supplies a finite open cover of the Picard scheme by families of effective divisors of a common degree. Their sum morphisms agree on the actual overlaps, since their twists are restrictions of the same Poin…
Over the field extension constructed from the curve's chart sections, the glued divisor sum factors every pointed morphism to a smooth proper geometrically integral group scheme. The equality is checked on geometric poin…
Actual chart sections give cohomology-vanishing opens which cover the represented Picard scheme. On each open, an actual effective divisor of a common positive degree has normalized line bundle equal to the Poincare rest…
Picard and Jacobian varieties of algebraic curves, including relative constructions, Albanese maps, autoduality and Weil pairings. Includes Kummer theory and torsion constructions.
Picard groups, degree-zero classes, torsion characters, and transport of supplied perfect pairings.
The Jacobian, its group object, and every Abel--Jacobi map use the same scheme constructed by finite faithfully flat descent without a rational basepoint. The genus is the geometric function-field genus, also identified…
The geometric generic Weil--Kummer identity determines the actual global derived-cohomology pullback. The auxiliary geometric model is supplied here with its germ and field-generation data; construction of this model is…
Finite H¹ pullback detects the actual abelianized fundamental-group map. Composing with multiplication-cover monodromy gives the actual full Tate module. The formula on the abelianization quotient records the inducing ma…
The geometric generic Weil--Kummer identity determines the actual global derived-cohomology pullback. The auxiliary geometric model is supplied here with its germ and field-generation data; construction of this model is…
This proof constructs the Kummer representative and the division divisor from the actual Abel--Jacobi multiplication fibre. It derives the fixed function, fixed divisor, evaluation and conorm identities before applying t…
A nonempty finite boundary has one independent sum relation. This counts the actual kernel of the sum homomorphism, not a group postulated to model étale cohomology.
When the target fundamental group is commutative, the actual scheme-induced map factors through the source's quotient by its closed commutator subgroup. The existing finite H¹ comparisons identify pullback with precompos…
Use the exact inverse image of abelian sheaves and its actual comparison with constant sheaves. In degree one, the resulting map sends an extension class to the class of its inverse-image extension, with the original coe…
The actual inverse-image map on derived cohomology corresponds to precomposition by the actual homomorphism of geometric fundamental groups. The proof uses the inverse-image extension, the map of its lift sheaf, and the…
The inverse-image adjunction unit sends a lift of the integer section one to a lift in the actual inverse-image extension. Compatibility with the quotient comes from the unit's naturality and the proved constant-section…
Divisibility of every valuation order supplies local expressions as a unit times an nth power. Compactness supplies the finite cover required by the concrete normalization theorem. The result is the actual normalization…
The explicit scaling automorphism of the generic root algebra extends through the universal property of relative normalization. The construction uses the original normalized finite étale cover, without assuming its conne…
The character comes from the proved root-scaling torsor structure on the original finite etale cover. The actual derived H¹ comparison then constructs a degree-one etale cohomology class. Its vanishing is equivalent to s…
At a geometric generic point the character on a product is computed using the product of the actual roots. Surjectivity of the generic Galois map detects equality of characters. Naturality of the same root-scaling automo…
A chosen root defines a point of the original generic root algebra and hence of its relative normalization. Root scaling and actual Galois monodromy send that point to the same point precisely at the usual root quotient.…
The automorphisms of the actual normalized Kummer cover act on its local root coordinate by multiplication with the root of unity pulled back from the base field. Equality of sections is proved using the schematic domina…
Every point of the original curve has an affine neighborhood on which the actual normalized cover represents roots of a unit. Its distinguished section is compared with the literal root of the original generic Kummer alg…
A constant normalization factor has an nth root over the algebraically closed ground field. Multiplying by that fixed root identifies the normalized evaluation root with the actual generic Kummer character.
An actual root of a Kummer function constructs a section of its normalized finite etale cover. Use a single open set with local denominator equal to that root: the normalized equation is then the unit equation, so the ex…
Local root coordinates on the original normalization identify its geometric fiber with nonzero roots of an actual unit. The root-scaling formula and uniqueness of roots up to a root of unity prove that the existing actio…
An arbitrary finite character of the scheme fundamental group determines an actual Kummer root in the separable closure of the function field. Its root field is the generic fiber of a finite etale cover. The orders of th…
The actual Kummer character is multiplicative and has exactly the subgroup of powers as its kernel. It therefore descends injectively through the proved divisorial Kummer quotient. The proved finite H¹ comparison then su…
Functions whose orders are divisible by a positive integer, modulo powers of that integer, identify with the actual torsion subgroup of the degree-zero divisor class group. The map sends a function to its principal divis…
Actual Kummer cohomology identifies finite-coefficient derived H¹ with divisor torsion. The actual Picard representation, the geometric dimension formula, and multiplication degree compute its cardinality.
The actual multiplication-cover monodromy is evaluated by the divisorial Weil pairing, in the torsion coordinates of the same Picard representative. These continuous roots-of-unity characters define classes in actual der…
Finite-exponent characters of the actual fundamental group factor through the corresponding multiplication cover. Geometric nondegeneracy and finite torsion duality then give all characters, not only an injective family.…
The evaluation formula supplied by existsjacobianweilcohomology identifies the character with the actual multiplication-cover action. Surjectivity of that action and geometrically proved nondegeneracy of the divisorial p…
The existing moving lemma chooses the divisor first. Torsion of its class then produces a genuine rational function with precisely the required divisor, so the representative avoids the actual Abel--Jacobi basepoint.
A residue vector of sum zero lifts to a degree-zero integral divisor supported on the boundary: subtract its total integral degree at a chosen boundary point. Actual Pic⁰ divisibility then realizes that divisor modulo th…
The degree of an actual principal divisor is zero. Orders away from the boundary vanish modulo the exponent, so the sum of its boundary residues is zero. Together with boundary realization this identifies the residue ima…
The normalization construction is applied to the actual open curve, using local unit equations proved from its interior valuation orders. The cover is not required to extend étale across the removed boundary.
The explicit scaling automorphism of the generic root algebra extends through the universal property of relative normalization. The construction uses the original normalized finite étale cover, without assuming its conne…
The character comes from the proved root-scaling torsor structure on the original finite etale cover. The actual derived H¹ comparison then constructs a degree-one etale cohomology class. Its vanishing is equivalent to s…
At a geometric generic point the character on a product is computed using the product of the actual roots. Surjectivity of the generic Galois map detects equality of characters. Naturality of the same root-scaling automo…
A chosen root defines a point of the original generic root algebra and hence of its relative normalization. Root scaling and actual Galois monodromy send that point to the same point precisely at the usual root quotient.…
The automorphisms of the actual normalized Kummer cover act on its local root coordinate by multiplication with the root of unity pulled back from the base field. Equality of sections is proved using the schematic domina…
Every point of the original curve has an affine neighborhood on which the actual normalized cover represents roots of a unit. Its distinguished section is compared with the literal root of the original generic Kummer alg…
A section of the actual normalized cover is equivalent to an actual nth power in the original function field. Consequently the kernel of the derived H¹ class is exactly the previously defined subgroup of powers.
Local root coordinates on the original normalization identify its geometric fiber with nonzero roots of an actual unit. The root-scaling formula and uniqueness of roots up to a root of unity prove that the existing actio…
The actual quotient of functions unramified away from the boundary by actual powers has n^(2g+r-1) elements for a nonempty finite boundary. The proof uses its exact residue sequence, actual Pic⁰ torsion, and the proved g…
Every character supplies an actual finite étale generic root cover. The local étale calculation proves divisibility of orders at all interior points. The original normalized Kummer cover recovers that character, so the a…
The constructed cover supplies the map; multiplication is checked on actual generic roots, and its kernel is exactly the actual subgroup of powers. Surjectivity requires the converse interior valuation calculation.
Divide the actual Pic⁰ class of a degree-zero divisor by the exponent. Subtracting the resulting multiple gives a principal divisor and hence an actual nonzero function with the prescribed orders modulo the exponent. The…
For a set of omitted places, a function is unramified away from the boundary when its orders there are divisible by the exponent. Boundary orders modulo the exponent define a homomorphism on functions modulo powers. Its…
The condition defining the divisorial open Kummer group is proved from the actual cover, using the actual local rings of the open curve.
Only the valuation at the point in question is used to remove an nth power and obtain a unit. For an actual open immersion, the canonical function-field map transports these two local-ring membership statements. No compa…
Weil reciprocity supplies the normalized root from actual division divisors. The semilinear division-difference formula then identifies its Weil pairing with the actual generic Kummer monodromy, rather than a separately…
Extension of constants on divisors induces a homomorphism on degree-zero divisor classes.
A ring equivalence between the constant fields transports places, divisors, and repartitions. The resulting quotient equivalences preserve genus and the Picard group.
The order of a degree-zero divisor class divides a principal multiple.
Commuting divisor correspondences induce commuting maps on the degree-zero Picard group.
Transport places, divisor classes, and scalar actions along the given field equivalence.
Picard group points and the Poincare bundle over an algebraically closed field.
Principal multiples of degree-zero divisor representatives define an additive map from torsion classes to differentials. Separability and a nonzero differential make this map injective; its value agrees with any principa…
A torsion class admits a degree-zero divisor representative whose multiple is principal. The logarithmic differential of a witnessing function is independent of this choice, is additive, and is injective under the stated…
Lifts places uniquely, extends divisors, and descends principal divisors through the corresponding spaces of rational functions to obtain an injective map on Picard groups.
A Jacobian scheme and its Picard-group parametrization over an algebraically closed field.
Torsion divisor classes admit rational functions with orders divisible by the torsion exponent.
Approximation of valuations moves a degree-zero divisor class away from finitely many places.
A smooth proper connected Jacobian presentation gives division by an integer that is nonzero in the base field. The proof constructs multiplication morphisms, computes their cotangent maps using square-zero extensions, a…
A torsion divisor class yields a rational function with the prescribed multiple of its divisor.
The relative group law on the Picard scheme of a curve model.
Divisibility of the Picard group over a finite extension of a rational function field.
A field ring equivalence respecting the constant field transports places, orders, and residue degrees. These comparisons carry a principal representative of a transported divisor back to the original divisor.
Weak approximation moves a divisor away from any prescribed finite set of places. The resulting differences with a fixed place generate the degree-zero Picard group.
Over algebraically closed constants, a divisor class vanishes precisely when its defining function is the prescribed power, assuming functions of zero divisor are constant.
The conorm on divisor classes is bijective on torsion over algebraically closed constant fields of characteristic zero, so their torsion groups have equal cardinality.
A principal natural multiple of a divisor annihilates its degree-zero Picard class.
Pushforward of degree-zero Picard classes commutes with intertwined semilinear actions.
Pullback followed by pushforward on the Picard group is multiplication by the extension degree.
The image of the torsion logarithmic differential map is the space of regular Cartier-fixed forms.
A surjective morphism has degree one. Its divisor pullback and pushforward are inverse, and conjugating field automorphisms yields the corresponding Picard equivariance.
Logarithmic differentials of torsion divisor classes carry correspondences to pullback and trace.
An injective map from Picard torsion to its character group is bijective under the finite-cardinality and roots-of-unity hypotheses.
Constructs the additive map from torsion divisor classes to characters whose evaluation agrees with the Weil pairing of any representing divisor data.
Represent a torsion Picard class by a divisor avoiding a prescribed finite set of places.
A principal integer multiple of a divisor annihilates its degree-zero Picard class.
The pullback of divisors descends to the degree-zero Picard group.
The pushforward of divisors descends to the degree-zero Picard group.
Base change of a relative Picard-zero designation and the resulting structure morphism and zero section.
The Picard comparison and its coefficient equivariance are constructed from the representing Picard scheme and the Abel--Jacobi point formula. A point of the actual multiplication fibre then supplies a moving divisor and…
Construct the Abel–Jacobi family from the line bundle defined by a moving point and the fixed basepoint.
The literal pullback lift retains its point-minus-basepoint formula in the base-changed Picard representation. The Poincaré comparison is supplied by existsrepresentsRelSubPicbaseChange. Pullback associativity supplies i…
For an abelian scheme over an algebraically closed field, this file constructs an invertible sheaf whose tensor powers have polynomial Euler characteristic. The coefficient in the relative dimension is positive.
An abelian scheme over an algebraically closed field admits an invertible sheaf that is finite by sections. The construction uses tensor powers of line bundles and the geometric properties of the abelian scheme.
The point comparison is constructed from the same Picard representation and Abel–Jacobi morphism over the original field. Coefficient automorphisms act through the actual pullback curve and its function field. Agreement…
Compare points, multiplication, and the Poincare bundle after changing the base ring.
The geometric points defined by the sections commute with the native pullback reassociation. Consequently the Riemann--Roch chart condition passes to every scalar extension, including field factors of a finite etale spli…
The established point-minus-basepoint formulas determine the comparison on every divisor class. The geometric base-change theorem identifies the actual lifted point with the unique valuation lift used by conorm. The germ…
Uniqueness of the actual lifted valuation implies that an action fixing the old function field fixes each lifted old place, and hence its conorm divisors. The fixed-function-field input is supplied by the geometric proje…
The curve supplies its Picard scheme and its actual group law. The proved geometric surjectivity of multiplication then supplies divisibility of Pic⁰; neither a Jacobian presentation nor divisibility is an input.
Combine closed-field Picard representability with the curve model point-class comparison.
Identify infinitesimal deformations with first cohomology and compute the dimension at zero.
Divisor presentations compute Euler characteristics on rational curve models. An invertible module with the Euler characteristic of the unit module is trivial; the corresponding pullback criterion is obtained from the sa…
The genus determines the relative dimension of the representing Picard scheme.
The tensor corrections for two curve isomorphisms combine into the correction for their composite. Classification of the resulting families proves the cocycle equation for the actual comparison morphisms over any commuta…
A relative Picard designation records a scheme over the base with a chosen zero section. The avatar bridge identifies an additive group with its points over a test scheme and records compatibility with the relative group…
Given the Picard point equivalence supplied by existspic0algPointsequivariantofabelJacobi, an actual division point has a divisor representative avoiding any prescribed finite set, together with a function realizing its…
Euler characteristics of section twists tensored with divisor ideals on geometric fibres.
Finite-map data on the original pointed curve produces finite etale chart sections. The resulting local representing scheme descends to the original curve. Over an arbitrary field the required finite-map data are constru…
Construct the Abel-Jacobi morphism using line bundles of relative points and rigidification.
Compare cohomology under the canonical fibre isomorphisms. Chart sections and the Riemann–Roch dimension calculation then provide a chart whose prescribed twist has vanishing H¹, first over algebraically closed fields an…
Construct finite presentations by sections for powers of the theta bundle over an algebraically closed field, using its translation law and stabilizer.
Split the finite etale algebra after a faithfully flat extension to obtain distinct sections. Riemann--Roch then selects the sections required by the chart criterion.
Place any finite set of points of a representing Picard scheme in an affine open.
Construct a finite morphism from a representing Picard scheme to a projective spectrum.
Compare the pullback of a fibre module tensored with a section twist along the specified fibre isomorphism.
Use a universal divisor and an auxiliary twist to construct a chart for fibrewise algebraically trivial rigidified line bundles. The vanishing of fibre H¹ identifies the chart locus, and uniqueness of divisors proves its…
Construct open charts covering the relative algebraically trivial Picard presheaf.
Find open chart ranges characterized by the vanishing of fibrewise first cohomology.
Identify degree-zero divisor classes with represented Picard points and compare Abel-Jacobi maps.
Identify degree-zero divisor classes with Picard points and compare nonzero sections with positivity of the Riemann--Roch dimension.
Construct the point-to-divisor-class equivalence from the represented Picard functor and compare pullback H0 with Riemann--Roch spaces after twisting by the base point.
Identify degree-zero Picard classes with the points of a representing Picard scheme, and compare this identification with an Abel–Jacobi morphism after field extension.
Compare restriction to a thickened section with fibre cohomology. The determinant of the evaluation map defines a theta section whose zero locus is detected by H0.
Relate vanishing of a theta section at a Picard point to nonzero sections of the corresponding Poincare bundle with a section twist.
Construct the divisor associated to an invertible module whose fibres have vanishing H¹ and one-dimensional H⁰. Its ideal is unique among divisors representing the same module up to tensoring by an invertible module pull…
Construct relative effective Cartier divisors representing a rigidified line bundle on the open locus where the prescribed twist has vanishing fibre cohomology. The construction compares pullbacks and cancels the auxilia…
Construct the group law and normalized Abel–Jacobi map from the represented Picard functor, and identify geometric-fibre point differences with divisor classes.
Open Picard charts glue to a smooth proper geometrically connected representing scheme.
Construct chart sections from Riemann--Roch and apply the finite-map representation theorem.
Transport the representing property and rigidified line bundles across a change of base.
Descend a relative Picard representation along a finite etale faithfully flat extension.
Construct the short exact pushforward sequence associated to an ideal of a section, using fibre cohomology vanishing.
Construct a nonzero theta section and prove that its translation stabilizer is trivial.
Construct the cohomology classification on two affine charts and prove its naturality.
Transport two-affine fibre covers through field extension while preserving the dimensions of their two cohomology groups.
Compare fibre zeroth cohomology and vanishing of first cohomology after extending the residue field.
Compare the cohomology of a fibre cover with the base change of its chart Čech differential to the residue field.
Base change the finite flat norm to each geometric fibre and transport a witnessing family through that norm. Rigidification changes the fibre only by a trivial line bundle.
Show that the relevant fibre line bundles have one-dimensional zeroth cohomology when their first cohomology vanishes.
The unit sheaf on a geometric fibre has a one-dimensional space of global sections.
Recover the genus from first cohomology of the unit sheaf using the chart Euler characteristic.
Transport geometric-fibre cohomology vanishing and dimension conditions along an isomorphism of module sheaves.
The genus here is fixed by the original curve and its actual geometric function field, independently of the choice of point, affine cover, or Picard representation. No assumption on the characteristic or algebraic closed…
Identify geometric-fibre invariants using chart sections and the function-field Riemann--Roch comparison.
Express an algebraically trivial invertible sheaf as a tensor product of differences of points and the chosen basepoint.
Algebraic triviality on a smooth proper curve is characterized by the Euler characteristic of an invertible module relative to that of the unit module.
Use a nonzero section and tensor comparisons to obtain an isomorphism with the unit module.
Pullback to the rational curve model is trivial and has Euler characteristic one.
Algebraic equivalence gives the same Euler characteristic as the unit module. Vanishing of its first cohomology produces a nonzero section of the pulled-back invertible module, and hence a trivialization on the proper sm…
Transport algebraic equivalence to zero along an isomorphism with a point-minus-basepoint bundle.
Transfer algebraic equivalence to zero along an isomorphism with a point-difference line bundle.
Show that the line bundle defined by a point minus the basepoint is algebraically equivalent to zero.
Transport algebraic triviality through a locally restricted norm and rigidification.
Tensor powers of an algebraically trivial invertible module remain algebraically trivial.
For a fibrewise algebraically trivial line bundle, cohomology detects the closed triviality locus.
Compare the dual-number cover maps and their induced first-cohomology classes.
Transition units of the form 1 + ε f glue an invertible module whose reduction and rigidifying section are trivial. Its class maps to the prescribed Cech class.
Prove finiteness of a relative Picard scheme when every geometric fibre has genus zero.
Invertibility of the theta bundle obtained from the determinant of the Picard bundle.
Descend line-bundle isomorphisms through finitely generated base algebras to prove the finite- presentation injectivity of the relative Picard presheaf.
Transfer finite-presentation injectivity from the relative Picard presheaf to its algebraically trivial cut.
Obtain finite-presentation surjectivity of the relative Picard presheaf from the line-bundle pullback construction.
Descend a rigidified line bundle to a finitely generated subalgebra, then enlarge that subalgebra so its fibres are algebraically trivial. Openness of the triviality locus and the affine limit comparison ensure that the…
Deduce local freeness of a pushforward line bundle from its fibre cohomology.
Use finite-type base change and fibre cohomology to obtain a pushforward module of constant rank.
This file studies a rigidified line bundle, tensored with a section twist, on the infinitesimal thickenings of a section of a proper smooth relative curve. Its pushforward is locally free of the thickness rank, and its d…
Prove that vanishing of first cohomology on every fibre is an open condition on the base.
On an affine base chart, compare the Euler characteristics of a line bundle and the unit module. Local constancy of these characteristics makes their equality locus open.
The universal effective-divisor construction supplies a proper surjective covering of the Picard scheme.
The diagonal is closed because triviality of the difference of two line bundles is a closed condition.
Finite-map data give the section comparison needed to specialize faithfully flat descent to finite étale base change over a reduced Noetherian base.
Prove the singleton sheaf condition by descending line bundles and their isomorphisms.
The algebraically trivial relative Picard presheaf satisfies Zariski descent when scalar extension induces bijections on global sections.
Finite-map data supplies the section comparison needed for Zariski descent.
The data records a relative Picard designation, its group law, and a point comparison.
For a curve with a two-affine-chart cover, this file descends a rigidified line bundle to a finitely generated subalgebra of the coefficient algebra. The affine-chart modules and their overlap maps are constructed at tha…
Finite presentation of the Picard functor yields finite-generation factorizations for affine points.
Identify the fibres of the Abel–Jacobi family with the point-difference line bundles.
Compare determinants of pushforwards before and after tensoring with the ideal of a section.
Identify the tensor product of graph-ideal inverse modules and basepoint factors with the point- difference line bundle.
Trivialize fibrewise algebraically trivial line bundles when the geometric fibres have genus zero.
The pullback comparison for theta bundles and the transport comparison over the extended base identify the two theta bundles under the stated fibre-cohomology hypotheses.
Compatibility of powers of the section ideal and their module sheaves with base change.
Compatibility of theta bundles with base change under the stated fibrewise cohomology hypotheses.
Compare pushforward and pullback of an invertible sheaf under the stated geometric-fibre cohomology hypotheses.
The section-twist and pushforward comparison isomorphisms identify Picard bundles under base change. Exterior powers and duality then identify the associated theta bundles.
Transport theta bundles along an isomorphism of the underlying rigidified line bundles.
Compare theta bundles after twisting by the difference of one point and the basepoint.
Iterate the theta-bundle comparison over a list of differences of points and the basepoint.
Prove the tensor relation for theta bundles twisted by two algebraically trivial line bundles.
Construct the tensor relation among translates of the theta bundle on a Picard scheme.
The fibre comparison pulls back the section ideal to the original section ideal. Compatibility with powers and inverse modules identifies the corresponding section twists.
The overlap unit and compatibility condition specify the Cech class of a rigidified deformation.
Nonvanishing of pullbacks of the adjunction counit from the prescribed geometric-fibre sections.
Identify the effective divisor ideal determined by a line-bundle isomorphism and fibrewise triviality.
A tensor comparison with an algebraically trivial line bundle preserves the divisor degree.
Recover an effective divisor by pullback from an isomorphism of its twist module.
Show that effective-divisor twist modules have algebraically trivial fibres.
The point ideal is invertible after restriction to the smooth open and is the unit ideal on the complement of the graph. These two descriptions give an invertible ideal globally.
The graph squares identify the pulled-back point ideal. Express each ideal module and inverse ideal module as the corresponding one-term tensor expression, then apply the existing pullback comparison for invertible ideal…
Representations of the sub-Picard presheaf determine relative Picard designations and their zero sections. A locally surjective family of representable open charts glues to a designation representing the presheaf through…
The represented group law and geometric connectedness give geometric integrality. The cotangent calculation at the rational zero section then determines the smooth relative dimension as the first-cohomology dimension on…
This file compares the Euler characteristic polynomials of tensor powers of an invertible sheaf and its pullback by a group-compatible endomorphism of an abelian scheme. In the relative dimension, pullback multiplies the…
For a group-compatible endomorphism of an abelian scheme whose kernel morphism is not finite, this file proves that the relative-dimension coefficient vanishes in the Euler characteristic polynomial of tensor powers of t…
On a commutative abelian scheme of relative dimension g over an algebraically closed field, multiplication by a natural number n has endomorphism degree n ^ (2 g) when the image of n in the field is nonzero.
Given a finite family of group-compatible endomorphisms of a commutative abelian scheme, this file constructs a homogeneous polynomial of degree twice the relative dimension. Evaluation at integer weights gives the endom…
This file constructs an invertible sheaf on an abelian scheme that is isomorphic to its pullback by group inversion. Its tensor powers have polynomial Euler characteristic with a positive coefficient in the relative dime…
For two group-compatible endomorphisms of a commutative abelian scheme, this file expresses the endomorphism degree of their integer combinations in the induced point group as the evaluation of a rational polynomial. Its…
A Poincaré family on a designated scheme classifies the line bundles satisfying a sub-Picard condition through unique maps over the base. Its restriction to the chosen zero section is trivial. The chosen classifying map…
Restrict a representing relative Picard designation along a base change.
Transport the group structure of a relative Picard presheaf to its representing designation.
Reduction-trivial points of a representing Picard scheme give rigidified dual-number deformations.
A representing relative Picard designation classifies the permitted rigidified line-bundle classes and determines a representation of their presheaf. Conversely, a presheaf representation compatible with the zero class s…
Classify pullback of the Poincare bundle to obtain the induced morphism of Picard schemes.
Compatibility of rigidified pullback classification with composition.
Compare classification of tensor products and units under the norm transform.
A transformation compatible with classification and tensor products preserves the represented multiplication. Compatibility with the unit bundle preserves the zero section.
Identify kernel points with deformation classes and compare multiplication and base change.
A multiplication-compatible morphism induces a unique additive endomorphism of the group of points after base change, characterized by its first projection.
A semilinear morphism preserving multiplication induces a unique additive map on points classified by the base-changed Poincare bundle, with the prescribed first projection.
A transformation compatible with isomorphisms and pullback induces a unique morphism between the schemes representing the two sub-Picard functors.
The pullback norm construction commutes with a compatible change of curve.
Construct the represented Picard endomorphism induced by pullback and the norm.
Rigidification of pullback preserves algebraic equivalence, tensor products and the unit, and is natural in the base. Representability therefore supplies the associated Picard morphism, its group compatibility, and its P…
A map of the curve over a split change of base induces a classifying map. The rigidified pullback construction preserves multiplication and the identity.
Twisting by the inverse of the restriction at a section constructs mutually inverse comparison morphisms between the corresponding represented Picard schemes.
Multiplication on two points factors through an automorphism of the fibre product and its second projection. This proves flatness from the flat structure morphism.
Kernel points of the representing Picard scheme correspond bijectively to rigidified line bundles whose pullback to the dual-number reduction is trivial.
The identification of infinitesimal kernel points with rigidified line bundles preserves multiplication.
Base change on infinitesimal Picard points is surjective, with fibres given by multiples.
Naturality of a relative group law makes its reduction kernel closed under multiplication. Every multiplicative endomorphism preserves that kernel.
The norm comparison for point divisors identifies the corresponding Picard multiplication.
Pullback identifies the line bundles and ideal modules of corresponding points. Combining these identifications with rigidification and the Poincare tensor product shows that the induced Picard map carries the specified…
The Poincaré norm comparison restricts to parameters whose image lies in the flat locus.
Compare the represented norm transformation and its Poincaré family after base change.
The Poincaré bundle comparison with rigidified norms is preserved by base change.
Compare the Poincare bundle along multiplication with the tensor product of its two pullbacks.
Triviality of the Poincare bundle pulled back along the identity section of the relative group law.
Pulling back a classified rigidified bundle along a field-valued point recovers the underlying pulled-back invertible module, using triviality of rigidification over a field.
Poincare line bundles under rigidified pullback and postcomposition.
Pullback along a power of a point agrees with the corresponding tensor power of the Poincare bundle.
Pullback on the representing Picard scheme agrees with rigidification of the pulled-back Poincare bundle on the changed curve.
Pulling the Poincare bundle back along multiplication gives the tensor product of its pullbacks.
The supplied norm comparisons commute with base change and tensor products. After rigidification, these comparisons give the tensor law for pullback along multiplication on the represented relative Picard group.
Pulling the Poincare bundle back along the identity section gives the unit line bundle.
Transport of Picard morphisms through commuting curve isomorphisms.
A tensor-compatible relation on rigidified line bundles makes the associated morphism between representing schemes preserve multiplication of points.
Transport of the curve automorphism is compatible with the base change comparison.
Pullback of rigidified line bundles commutes with the comparison back to the base ring. The represented pullback morphism therefore commutes with the first projection of base change.
Composition and identity comparisons for curve pullback show that the induced morphisms of the represented relative Picard functors are mutually inverse.
The comparison between the Poincare bundles transports tensor products along base change. It identifies the multiplication represented after base change with the base-changed group law.
The norm map kills tangent points under the characteristic-rank hypothesis.
A morphism classified by rigidified norms preserves multiplication and the zero section.
Tensor compatibility of the rigidified pullback transformation yields multiplication compatibility.
Pullback morphisms commute with the classified transport of rigidified line bundles.
Transport points along the pullback defining the base-changed Picard carrier. Compatibility with multiplication and the identity gives the unique lift in the dual-number kernel.
The tensor inverses chosen for two successive changes combine to a tensor inverse for the direct change. Their evaluation maps give the required line-bundle isomorphism.
Compatible invertible sheaves on a finite faithfully flat base change descend, with their rigidifications, using the bijectivity of global sections.
Normalized overlap isomorphisms satisfy the diagonal and cocycle laws by rigidity. Effective descent for invertible module sheaves then gives the descended rigidified bundle.
The group law, its product and a rigidified line bundle descend together to a finitely generated subalgebra, with the cube pullback comparisons preserved.
Pullback comparisons and descent glue compatible rigidified line bundles over an open cover.
Noetherian affine charts representing triviality of a rigidified line bundle. The construction compares local triviality with the proper flat representability theorem.
The affine charts glue to the closed locus where every pullback of the rigidified bundle is isomorphic to the unit bundle.
Closed immersion charts for the triviality locus of a rigidified line bundle glue globally. Their kernels agree on affine overlaps and define the required closed subscheme.
Descend to a noetherian base and transport its closed triviality locus to each affine chart.
A descended closed immersion representing trivializations pulls back to a chart for the original rigidified line bundle. Noetherianity gives finite presentation.
Local finite-presentation criteria for the triviality locus of a rigidified line bundle.
Glue the closed loci on affine charts using the existing affine-chart criterion.
Descend the group law and the three face trivializations through finite generated models.
A rigidified line bundle and its group-law model descend to a finitely generated base over the integers on each affine open of the parameter scheme.
Noetherian descent of rigidified line bundles on affine opens. Finite subalgebra stages descend the group law, the invertible module and its rigidification.
Lift rigidified line bundles across a square-zero extension using a two-affine open cover.
A base-change square compatible with multiplication also preserves the unit section. The induced map on triple products therefore pulls back both the line bundle and its rigidification along the cube section.
Fibrewise algebraic triviality of a rigidified line bundle represented by points minus the base point.
Descend fibrewise algebraic triviality of rigidified line bundles from an open cover of the base.
Lift geometric points through a finite faithfully flat morphism to descend algebraic equivalence.
Apply affine descent to the finite faithfully flat base change.
Compatible isomorphisms after affine faithfully flat pullback descend to the original rigidified line bundles, using the bijectivity of global sections.
Pullback carries a trivialization to every member of an open cover. Conversely, the section comparison for the abelian scheme glues the local trivializations.
Descend to a noetherian model and pull the resulting trivialization back to the original bundle.
Infinitesimal lifting makes the closed triviality locus open, then connected fibres make it all.
Construct a unit isomorphism of rigidified line bundles over a square-zero thickening from the given pullback isomorphism.
Over a field, rigidifying an invertible module preserves its underlying line bundle up to isomorphism.
Triviality of a rigidified line bundle lifts across a small extension when its two unit faces are locally trivial over the base.
Use the naturality of the group identity to compare the identity face with base change. Local triviality on the face and the sections of the proper abelian scheme yield a global trivialization compatible with the rigidif…
Triviality along a rigidifying section after pulling back a rigidified line bundle from a field.
Show that the type of line bundle isomorphisms respecting fixed rigidifications is a subsingleton.
A trivial reduction lifts across the square-zero kernel of the dual-number projection.
On a smooth proper geometrically connected commutative group scheme over an algebraically closed field, the tensor product of the translates of a line bundle by two rational points is isomorphic to the translate by their…
This file proves a theorem of the cube for an invertible sheaf on a triple product over an algebraically closed field. Under the stated properness and integrality hypotheses, trivial pullbacks to the three fixed-coordina…
For a triple product whose first factor is a proper smooth integral curve, this file proves that an invertible sheaf is trivial when its three fixed-coordinate pullbacks are trivial. The other factors satisfy the propern…
For three maps into a smooth proper geometrically connected commutative group scheme, this file proves the tensor identity relating pullbacks of a line bundle along the triple product, the individual maps, and their pair…
For an invertible sheaf invariant up to isomorphism under group inversion, this file identifies its pullback by multiplication by an integer n with its tensor power of exponent n n. The group scheme is smooth, proper, ge…
The tensor-inverse evaluations identify changing the section before and after scalar extension. The actual Poincare families then identify the comparison morphisms. All base rings are arbitrary commutative rings, includi…
Choose a field factor of the actual finite etale splitting algebra, and transport the base-changed sections through the native iterated-pullback isomorphism. The Riemann--Roch condition is preserved by equality of the ac…
This file proves preservation of a specified short exact sequence supported on an infinitesimal thickening of a section of a proper smooth relative curve. The comparison map is required to agree with the original adjunct…
Smoothness of a relative Picard scheme representing the algebraically trivial subfunctor, using finite-map data.
Compute fibre cohomology of sufficiently positive twists of the Poincare bundle.
Compute cohomology of Poincare-bundle fibres over algebraically closed field-valued points.
Transfer vanishing of first cohomology from a residue-field chart to the corresponding fibre.
A universal effective divisor gives a surjective map to the represented degree-zero Picard functor.
The Picard bundle pushes a tensor product of a rigidified line bundle and a module forward to the test scheme. The theta bundle is the dual of the determinant of this pushforward after twisting by the chosen relative sec…
The Picard point equivalence is constructed over the literal coefficient map used by AlgPoints. Its additive refinement retains the actual Abel--Jacobi point formula. Restricting it to torsion then preserves the underlyi…
The cardinality is computed from the finite étale multiplication kernel and the geometric degree of multiplication. No cardinality or rank hypothesis is imposed on the group of algebraic points.
Every basepoint gives an actual point-minus-basepoint classifying morphism into the same Jacobian constructed without a rational point. The fixed comparison of Picard representations transports both the morphism and its…
The actual Albanese factorization for the fixed, unpointed Jacobian.
A point is required only after the finite field extension. Normalization there and the unrigidified base-change comparison make the actual Picard representation a representation of a restricted presheaf. Its canonical de…
The same descended scheme is used throughout. Its locally constant smooth dimension is compared, after the actual finite extension, with the dimension of the represented Picard scheme. Comparison of affine smooth charts…
A closed point supplies a finite residue extension and an actual section of the extended curve. Pointed Picard existence and its affine-neighborhood theorem apply there. The unrigidified quotient presheaf supplies the de…
The group law comes from the actual tensor-product group of unrigidified classes, using the proved equivalence between descended scheme points and matching local classes. No group law or comparison is assumed on the desc…
A single effective descent presentation is chosen from the proved existence theorem. Its carrier and structure map are retained by all public constructions. The relative group law is constructed from tensor product, and…
When an actual rational point and a pointed representation are provided, the same chosen unpointed Jacobian represents the unrigidified presheaf. Descending the local universal element fixes its comparison with the point…
The comparison uses the descended universal element. Its effect on the identity and tensor products is determined by the actual classifying equivalences.
Classified transport along a curve isomorphism compares the two Abel–Jacobi morphisms.
Restriction maps on Picard points commute with a compatible base change. The proof compares the pulled-back Poincare bundles using explicit pullback isomorphisms.
The comparison of Poincaré bundles under base change identifies the multiplication laws on points of the two representing schemes.
A section disjoint from the support of an ideal pulls back to the unit ideal on its subscheme. The resulting empty subscheme has finite rank zero.
Euler characteristic of a supported divisor line bundle tensored with an ideal module.
Euler characteristic of the section twist tensored with a supported ideal module.
Euler characteristic after pulling back a fibrewise line bundle and its divisor twists.
Euler characteristic of section and ideal twists on the first rational component.
Euler characteristics of finite tensor products of positive and negative section twists. Tensor powers contribute their signed fibre degrees, and list induction adds the contributions.
The iterated base-change fibre is isomorphic to the original fibre, compatibly with both projections.
The Euler characteristic of fibres is well-defined and has clopen level sets.
Finite etale splitting and Riemann-Roch supply the required families of chart sections.
Lift a basis of the structure-sheaf pushforward through a cartesian dual-number thickening. The proof uses the unique constant and infinitesimal decomposition on affine opens.
A proper flat family with one-dimensional geometric global sections has constant first-cohomology rank.
Translate the Riemann--Roch point-selection argument through the curve model and Cartier ideals.
Universal supported divisors provide affine neighbourhoods of finite sets in the charts. The section and divisor ideals are invertible, and section twists commute with the given base changes inside the relative smooth lo…
Compatible frame systems determine Cech classes independently of the chosen frames. The construction descends those classes through the rigidified kernel quotient.
Construct chart frames for the norm of a line bundle, and compute their infinitesimal transition using the trace on the original overlap algebra.
Supported Cartier divisors and the open cohomology-vanishing locus construct a chart with the required factorization and uniqueness properties.
The vanishing of first cohomology and the support of zero schemes cut out an open parameter space. Universal effective divisors give its chart and its lifting property.
Assemble the polarised Cartier charts using cohomology and support conditions.
Assemble the finite family of Cartier charts and prove local surjectivity.
The open locus where first fibre cohomology vanishes.
Fibre cohomology of dimension one provides local generating sections. Their zero-scheme support conditions glue to an open locus in the parameter scheme.
Nonzero sections of the twisted chart module cut out supported Cartier divisors of the prescribed degree on smooth geometric fibres.
Constructing the supported Cartier divisor of a fibrewise section.
This file constructs a relative Jacobian for a proper smooth geometrically integral relative curve with a section over a discrete valuation ring. The resulting abelian scheme has an Abel-Jacobi map normalized at the sect…
This file constructs the relative Jacobian of a proper smooth relative curve with a section, using finite maps of arbitrarily large degree with generically etale level sets. Over a discrete valuation ring, the constructi…
Constructing the representing Picard scheme from the specified chart data.
A finite etale extension supplies chart sections for the curve. The represented Picard scheme over that extension is then descended, together with its smoothness, properness and geometric connectedness.
Representations over a covering family of localizations glue to a global Picard representation.
Construct the representing scheme from finitely many open charts and bijective base sections.
The sheaf property and affine orbit neighbourhoods descend the representing scheme. The represented unit defines its zero section, while the given smoothness, separation, quasi-compactness and geometric connectedness des…
Split finite algebras give injective families of sections in the smooth locus. Choose subsets on each side, add their effective Cartier divisors, and transport the total degree to obtain finite two-sided chart data with…
The closed fibre and the generic Poincare comparison give algebraic triviality on every fibre.
Rank one of fibre sections from vanishing first cohomology and the supported twist condition.
Rank one of fibre sections for a supported divisor line bundle.
A bijective scalar map identifies the global sections of the fibre with its field. Transport along the identity-base-change isomorphism and the two-chart comparison then computes the dimension of the zeroth cohomology of…
The canonical pullback comparison transports two-affine-open covers and identifies the zeroth and first cohomology of the associated fibre modules in both directions.
Geometric connectedness of the scheme representing algebraically trivial line bundles. Algebraic equivalence connects closed points to zero, and algebraic closure descent handles fields.
Algebraic equivalence to zero on a smooth fibre is independent of the chosen algebraically closed point over the same point of the parameter scheme.
A filtered tensor product of positive and negative point ideals is algebraically trivial when its signed degree sum vanishes. The proof computes its Euler characteristic on a two-chart cover.
A family connecting two line bundles remains a family after tensoring with a fixed line bundle. A trivialization at the first point gives algebraic equivalence to zero.
Closedness of the fibrewise triviality locus in a flat family.
Upper semicontinuity of fibre H0 dimensions for an invertible sheaf.
Upper semicontinuity of fibre H1 dimensions for an invertible sheaf.
The supported section-ideal intersection is finite of the specified fibre rank.
A section supported in the smooth locus has an invertible ideal. Tensor its positive twist and negative ideal power, then induct on the finite list of sections to preserve invertibility and the specified pullback isomorp…
If a subscheme ideal vanishes near the section, the pulled-back section ideal is invertible and its zero scheme is finite of rank one over the field.
Compare an iterated fibre with the original fibre. Pull back the section ideal and a relative Cartier divisor, then use their support conditions to extend invertibility from the given open part of the curve model.
Fibrewise cohomology conditions give the pushforward base-change isomorphism.
The algebraically trivial sub-Picard presheaf inherits injectivity along filtered affine limits.
Openness of algebraic equivalence to zero gives local finite presentation surjectivity.
For an invertible module over a finite-type base change of a proper flat curve, finiteness of the two-chart Cech complex reduces local freeness of its pushforward to vanishing of fibre H1 and the prescribed dimension of…
Restrict to affine opens of the base, apply the finite-type rank theorem, and transport the fibre cohomology along the pullback comparison. Vanishing in first cohomology and constant zeroth rank give a locally free pushf…
Fibrewise vanishing of first cohomology defines an open locus.
Openness of algebraic equivalence to zero from a closed stratum and its complement.
Finite maps of arbitrarily large degree and the fibrewise Riemann--Roch bound show that the represented Picard scheme remains proper and geometrically connected after base change to a field.
Separatedness of the representing scheme from the closed fibrewise triviality locus.
Norms of infinitesimally trivial line bundles are trivial when rank equals the characteristic.
Pullback comparisons identify the Poincare bundle with the point and basepoint divisor modules.
A section twist and an effective Cartier divisor module become trivial away from their supports. Tensoring them preserves a given trivialisation on the fibre.
Cohomology comparison and the vanishing of the rational curve genus descend the triviality of an algebraically trivial pullback.
The section ideal is invertible when the section lies in the relative smooth curve locus. Its powers commute with base change, and the invertible-ideal comparison identifies the associated pulled-back modules.
The ideal of a section commutes with base change. Its powers and inverse modules therefore give the comparison isomorphism for section twists along a base morphism.
The fibre comparison homeomorphism carries the section point to the base-changed section. It transports the assumed containment in a connected component to the base-changed fibre.
The fibre comparison preserves the section point and the inverse image of the chosen open. It transports containment outside the section's component on non-smooth fibres.
Nonvanishing of the fibre pullback of the adjunction counit.
Equality of supported Cartier ideals from line bundle comparisons.
The existing section uniqueness and fibre cohomology assumptions identify a relative Cartier divisor after change of base. The proof transports the given bundle comparisons and retains the original support and zero-schem…
The existing section uniqueness and fibre cohomology assumptions identify a relative Cartier divisor after change of base. The proof transports the given bundle comparisons and retains the original support and zero-schem…
A group law is determined by its multiplication. The multiplication comparison for base change therefore identifies the represented relative Picard group laws.
Smoothness makes the representing scheme reduced, so the reduced geometric-point extensionality theorem identifies morphisms agreeing on algebraic-closure-valued points.
Geometric points of the generic fibre are dense in a flat scheme of finite type. For a reduced separated scheme this density detects equality of endomorphisms, including over a localization of the rational integers.
Square-zero lifting of rigidified bundles supplies formal smoothness on affine charts.
Vanishing of first cohomology after a field extension implies vanishing over the original field, using the comparison of two-affine-cover cohomology.
Faithfully flat pullback preserves nonzero sections of invertible modules. This descends a support bound from an extension of the residue field.
The zero section makes the structure map of the representing Picard scheme surjective.
The divisibility input in the existing Kummer detection proof is supplied by the proved geometric Jacobian construction and multiplication surjectivity, for every finite rational-function-field extension. No divisibility…
The actual unique valuation lifts define conorm on divisors and Picard groups. Transporting a Weil datum along those lifts preserves its pairing through the coefficient embedding. Hence the constructed divisorial Weil pa…
Divisorial data for the Weil pairing, with disjoint rational support, symmetry and addition.
The ratio of evaluation roots is the actual divisorial Weil pairing. For a semilinear automorphism this proves the orientation σ(β) / β directly.
Construct torsion classes from disjoint divisor data and prove the pairing laws from divisor evaluation and Weil reciprocity. A moving property provides simultaneous representatives; perfection is separately supplied whe…
Move divisor representatives apart and compare evaluation after pullback and norm.
Compare pairings at two torsion levels by using the same disjoint divisors and powering the witness function at the upper level.
Disjoint divisor representatives and the existing separable evaluation formula identify the pairing of a pullback with the pairing against the norm class.
Disjoint divisor representatives and the existing inseparable evaluation formula identify the pairing of a pullback with the pairing against the norm class.
Transport rational places, residue evaluations, and divisor witnesses along a semilinear automorphism to compare the resulting Weil pairing values.
Combine the proved injectivity with finite torsion duality. The divisibility hypothesis is retained exactly as supplied by the original theorem.
Injectivity of the divisorial Weil pairing from divisibility over finite function-field extensions. The proof uses Kummer covers and Weil reciprocity to detect each nonzero torsion class.
Use a nontrivial Kummer automorphism and moved divisors to detect every nonzero torsion class. Norm evaluation and Weil reciprocity give injectivity of the pairing map.
Bound the support of valuations by the coefficients of minimal polynomials. The resulting divisor pushes forward to the norm divisor, so its degree is zero.
The Weil pairing is unchanged when the defining divisors change by principal divisors.
Compare the two divisor evaluations using pullback and the field norm. The original algebra structure along the field homomorphism supplies each comparison.
Compare the two divisor evaluations using pullback and the field norm. The original algebra structure along the field homomorphism supplies each comparison.
Disjoint rational supports make both divisor evaluations, and hence the Weil pairing, nonzero.
Interchanging the two divisors in Weil pairing data inverts the pairing.