Algebraic variety

An algebraic variety is a geometric space locally described by polynomial equations, together with its regular functions.

A rational map is a morphism of algebraic varieties defined on a dense open subset, with representatives identified when they agree on a smaller dense open subset. For an integral source and a separated variety as target, agreement on a dense open determines a morphism uniquely. This definition also permits nonprojective sources.

For a possibly reducible algebraic variety $X$, take pairs $(U,f)$ with $U$ a dense open subset and $f$ a regular function on $U$. Identify pairs agreeing on a common dense open subset. Addition and multiplication on intersections make these classes a ring. For an affine variety with reduced coordinate ring $A$, this ring is its total quotient ring. For an irreducible variety, it is the function field. An isomorphism of dense open subsets induces an isomorphism of these rings.

Let $Y$ be the reduced union of the irreducible components of an affine variety $X$ containing $P$. Removing all other components leaves an open neighbourhood of $P$ on which $X$ and $Y$ agree, so their local rings at $P$ agree. In the coordinate ring $B=k[Y]$, every denominator not vanishing at $P$ avoids all minimal primes: each component of $Y$ contains $P$. Such a denominator is a non-zero-divisor. Consequently $B_{{\mathfrak{m}}_P}$ injects into the total quotient ring $Q(B)=\mathrm{Rat}(Y)$. One cannot generally replace $Y$ by a single component through an intersection point.

On an integral algebraic variety, a rational function is an equivalence class of regular functions on nonempty open subsets, with equality on a common dense open subset. These form the function field $k(X)$. On an affine open $\mathrm{Spec}A$, this field is the fraction field of $A$.

// Target: geometry-and-topology.bigb

A locally closed affine algebraic set in a projective space, equivalently an open subset of a closed subset of projective space cut out by homogeneous polynomials. Reducibility is allowed for an algebraic set; an irreducible variety with such an embedding is a quasi-projective variety.

An irreducible variety which admits a locally closed embedding into projective space. Open subsets of affine varieties and blowups of algebraic varieties along closed centres give important examples.

An algebraic variety whose underlying topological space with its Zariski topology is an irreducible topological space: it is nonempty and cannot be a union of two proper closed subsets. For a reduced affine variety, this is equivalent to its coordinate ring being an integral domain.

A constructible subset is a finite union of locally closed subsets in the Zariski topology. A dense constructible subset contains a dense open subset. Constructibility alone is weaker than closedness, but a constructible subgroup is closed.

The image of a morphism of algebraic varieties is constructible. This replaces the generally false assertion that all morphisms have closed image. Group structure adds the fact that a constructible subgroup is closed.

For an algebraic variety $X$ defined over a field $K$, a K-rational point is a morphism $\mathrm{Spec}K\to X$ over $K$. In affine coordinates it is a solution of the defining equations with every coordinate in $K$; in projective coordinates it is a nonzero coordinate vector over $K$, considered up to common nonzero scaling. The set is denoted $X(K)$. Over a number field, having points over every completion need not imply having a rational point; for genus-one coverings this failure is recorded by the Tate–Shafarevich group.

An algebraic variety is smooth over its ground field when its structural morphism is smooth. Over an algebraically closed field its local rings are regular; on an irreducible variety its tangent spaces all have the variety's dimension of a scheme.

Over a perfect field, a point of an algebraic variety is smooth exactly when its local ring is a regular local ring. The smooth locus of a variety is open; its closed complement is the singular locus.

The normalization of an integral algebraic variety is obtained by taking the integral closure of its affine coordinate rings in its function field. Its natural morphism is finite and birational, and its source is normal. For Cartier divisors, bigness under finite normalization compares section growth on the two varieties.

A closed subvariety is an integral closed subscheme of an algebraic variety. It supplies the lower-dimensional loci on which positivity is tested by the Nakai–Moishezon criterion.

An algebraic curve is a one-dimensional algebraic variety. On a smooth algebraic curve, the local ring at every closed point is a discrete valuation ring, which allows orders of zeros and poles to be recorded as a divisor on an algebraic curve.

For $a>0$, this algebraic curve can be constructed using a circle of radius $a$ centred at $(a,0)$ and its tangent line $x=2a$. On a ray from the origin, subtract the distance to its nonzero circle intersection from the distance to the line. The resulting radial distance is $r=2a{\sin }^2\theta /\cos \theta$, for $-\pi /2<\theta <\pi /2$. Thus a parametric curve for it isEliminating $t$ gives the displayed cubic equation. It has a cusp at the origin, horizontal tangent direction there, and vertical asymptote $x=2a$. Its name honours Diocles.

An algebraic curve which is projective, reduced and irreducible. Over an algebraically closed field, its only global regular functions are constants and its arithmetic genus is $h^1({\mathcal{O}}_C)\ge 0$.

For an affine plane algebraic curve $F(x,y)=0$, a point over an algebraic closure is nonsingular when the displayed condition holds. Its Zariski tangent space has dimension one, and a local coordinate can be chosen in the direction of a nonzero partial derivative. A smooth algebraic curve has all points nonsingular. Over a complete local field, the Hensel lemma lifts a nonsingular residue point by solving for a coordinate with unit partial derivative.

For a projective plane curve $C=V(F)$ and a nonsingular projective point $p$, the tangent is the line $F_x(p)x+F_y(p)y+F_z(p)z=0$. The coefficients are not all zero, and the Euler theorem for homogeneous functions makes this line pass through $p$. On an affine chart this is precisely the first-order part of a local defining equation, so the definition agrees with the usual tangent line and is invariant under projective transformations.

An affine algebraic curve is an affine variety of Krull dimension one. An irreducible affine curve has a finitely generated one-dimensional domain as its coordinate ring. Quotienting this ring by a nonzero element gives a zero-dimensional Noetherian ring, hence an Artinian ring. Consequently a nonzero regular function has finitely many zeros. No nonsingularity assumption is part of the definition.

For nonzero real $a$, the folium of Descartes is the algebraic curve $x^3+y^3=3axy$, with rational parametrization of an algebraic curveIt is symmetric about $y=x$ and has a node at the origin, with the coordinate axes as tangents. For $a>0$, $0\le t<\infty$ traces a loop in the first quadrant, returning to the origin as $t\to \infty$. Its enclosed area, by Green's theorem, isThe other branches approach the asymptote $x+y+a=0$ as $t\to -1$. Negative $a$ rotates the positive-$a$ picture through $\pi$.

The projective algebraic curve $D=\{y^{2}z=x^3\}\subset {\mathbb{P}}^2$ is an irreducible cubic with a cusp at $[0:0:1]$. In the chart $z=1$, its equation is $y^2=x^3$, with parametrization $(x,y)=(t^2,t^3)$. Its affine version $y^2=z^3$, used below, differs only by renaming a coordinate. At the point $[0:1:0]$, the chart $y=1$ gives $z=x^3$, which is smooth and exhibits its unique smooth flex of a cuspidal cubic.

Parametrize the affine cuspidal cubic $x^2=y^3$ by $x=t^3,y=t^2$. Removing the zero set of $g=x-y$ removes exactly the parameter values zero and one. Its regular-function ring is the localization $k[t^2,t^3{]}_g$. To recover the normalization coordinate inside this ring, use $g^2=y(y^2-2x+y)$: inverting $g$ makes $y$ a unit, so $t=x/y$ lies in the localization. As $g=t^2(t-1)$ is a unit, so are $t$ and $t-1$, proving the displayed equality. This calculation is valid in every characteristic. At the smooth point $(1,1)$ of the original curve the local ring is instead $k[t{]}_{(t-1)}$: $y$ is a unit there, and every polynomial in $t$ nonzero at one is invertible. That point is absent from the displayed open subset, so its local ring is computed in the original curve.

For $D=\{y^{2}z=x^3\}$, the determinant of the homogeneous Hessian matrix is $24x{y}^2$. Its intersections with $D$ are the cusp $[0:0:1]$ and the smooth point $[0:1:0]$. The Hessian criterion for a flex therefore gives a unique smooth flex, with tangent $z=0$ and contact order three. Consequently every pair consisting of a triple line and a cuspidal cubic meeting only at a smooth point is projectively equivalent to $(3\{z=0\},D)$.

The algebraic set in ${\mathbb{A}}^3$ defined byis the disjoint union of the cuspidal cubic $V(x,y^2-z^3)$ and the two reduced points $(-1,0,0)$ and $(1,0,-1)$. Its radical vanishing ideal is the intersection of the three corresponding prime ideals.

The scheme $C^{(d)}$ parametrizes effective degree-$d$ Cartier divisors on a smooth algebraic curve. Near a Cartier divisor $\sum m_i{P}_i$, its completed local ring is a power-series ring in the elementary symmetric coordinates of $m_i$ local parameters at each $P_i$. It is therefore smooth of dimension $d$, even when the characteristic of a field divides $d!$. For projective $C$ it is projective.

Over a field $k$, an affine curve parametrized by positive integer powers of one parameter. Its coordinate ring is the subalgebra generated by those powers, and its defining prime ideal is the kernel of $k[x_1,\dots ,x_n]\to k[t]$ sending $x_i$ to $t^{a_i}$. Equal exponent sums yield binomial relations. Studying those relations connects an additive semigroup of exponents to the geometry of the curve.

The kernel of $x\mapsto t^3$, $y\mapsto t^4$, $z\mapsto t^5$ is generated by $y^2-xz$, $yz-x^3$, $z^2-x^{2}y$. Reducing these three relations leaves a unique normal form $A_0(x)+A_1(x)y+A_2(x)z$: its images have distinct exponents modulo three. The curve ring is therefore finite free of rank three over $k[x]$ and has dimension one. In a three-variable polynomial ring its ideal has height two, but its three independent quadratic initial forms force at least three generators. This is a concrete failure of generation by codimension many equations.

A rational parametrization expresses the coordinates of an algebraic curve as rational functions of a parameter, with a rational inverse on a dense open subset. It identifies the function field with $K(t)$ and the smooth projective normalization of an algebraic curve with the projective line. A nonconstant parametrizing map without an inverse is a weaker notion; even this cannot exist for an elliptic curve in characteristic zero.

The arithmetic genus of a proper curve $C$ is $p_a(C)=1-\chi (C,{\mathcal{O}}_C)$. For an integral projective curve over an algebraically closed field, $H^0(C,{\mathcal{O}}_C)=k$, so $p_a=h^1\ge 0$. Its finite normalization has $p_a(C)=g(C^{\nu })+\mathrm{length}({\nu }_{\ast }{\mathcal{O}}_{C^{\nu }}/{\mathcal{O}}_C)$. Hence arithmetic genus zero forces the curve to be smooth and rational.

For a projective complete intersection curve cut out by a homogeneous regular sequence of positive degrees $a,b$ in ${\mathbb{P}}_k^3$, the Koszul resolution gives Hilbert polynomial $abm+ab(4-a-b)/2$. Hence its arithmetic genus is $1+ab(a+b-4)/2$. The same resolution shows $H^0(C,{\mathcal{O}}_C)=k$, and the dimension of a scheme is one, so this genus equals ${\dim }_k{H}^1(C,{\mathcal{O}}_C)$. Smoothness and irreducibility of the curve are not required. For degrees five and seven, the value is $141$.

In the classical convention over an algebraically closed field $k$, a prevariety is an irreducible topological space with a sheaf of rings which has a finite open cover by affine varieties, with compatible local identifications. A prevariety need not be separated. A classical algebraic variety is a separated prevariety: its diagonal morphism has closed image. Some authors allow reducible prevarieties and varieties; the separation criterion and product construction remain the same under that convention.

A group variety is an algebraic variety whose multiplication and inversion are morphisms.

An isogeny is a surjective morphism of algebraic groups with finite kernel. For abelian varieties it is a finite surjective group morphism. An isogeny of elliptic curves is its one-dimensional abelian-variety specialization.

The multiplicative group scheme represents units: its $R$-points are $R^{\times }$. Multiplication is specified by $t\mapsto t\otimes t$ on the coordinate ring. It is the principal open subset $D(t)$ of the affine line and the one-dimensional split algebraic torus.

// Target: ringed-space.bigb

The multiplicative algebraic group has ${\mathbb{G}}_m(K)=K^{\times }$, with multiplication as its group operation. Its coordinate ring is $K[z,z^{-1}]$. The formal multiplicative group describes its law near the identity $z=1$.

This is a group action whose action map $G\times V\to V$ is a morphism of varieties. Orbits are locally closed: constructibility gives a relatively open piece, and its translates cover the orbit. An orbit of an irreducible group is irreducible and open in its closure. These facts control degenerations of quiver representations.

The kernel is a closed identity fiber. The image is closed because a constructible subgroup is closed. Every nonempty fiber is a kernel translate, so the fiber dimension theorem gives $\dim G=\dim \ker \phi +\dim \mathrm{im}\phi$, even for inseparable morphisms.

Let $S$ be constructible in an algebraic group, and $C=\overline{S}$. The closure is a subgroup. Choose a dense open $U\subseteq C$ contained in $S$. For every $c\in C$, the dense opens $U,cU$ intersect, giving $u=cv$ for $u,v\in S$. Hence $c=u{v}^{-1}\in S$ and $S=C$. The Chevalley constructibility theorem therefore proves closedness of homomorphism images.

A homomorphism of group varieties is a morphism of algebraic varieties that also preserves multiplication. It automatically preserves the identity and inversion.

An abelian variety is a complete connected group variety. Despite its name, commutativity follows from completeness and the rigidity lemma.

If $N$ is invertible in the base field, multiplication by $N$ on an abelian variety is finite étale: its differential is $N$ times the identity, and properness turns its zero-dimensional fibres into finite fibres. The Theorem of the Cube gives $[N{]}^{\ast }L\cong L^{N^2}$ for a symmetric ample line bundle. Top intersection products give degree $N^{2\dim A}$. For each prime power ${\ell }^r\mid N$, the kernels killed by ${\ell }^s$ have orders ${\ell }^{2s\dim A}$; the classification of finite abelian groups forces $2\dim A$ cyclic factors of length $r$. Combining the prime powers gives the displayed group.

The Jacobian variety of a smooth projective algebraic curve parametrizes degree-zero line bundles. Its dimension is the genus of the algebraic curve. An effective degree-$d$ Cartier divisor defines a point in a degree-$d$ torsor through the Abel map of an algebraic curve. The theta divisor induces an isomorphism with the dual abelian variety.

A smooth projective curve over the complex numbers is determined up to isomorphism by its Jacobian variety together with its canonical principal polarization of a complex torus. For genus at least three, the branch locus of the theta Gauss map recovers the canonical curve by projective biduality theorem, or recovers the canonical rational normal curve and the branch points of a hyperelliptic curve. In genus two the theta divisor itself is the curve; in genus one the Jacobian variety is the curve after choosing an origin. The polarization is essential data in this statement.

For a smooth connected projective curve $C$ with chosen basepoint $p$, the Abel map of an algebraic curve $i_p:C\to J(C)$ sends $q$ to $[q-p]$. Every morphism $g:C\to A$ to an abelian variety with $g(p)=0$ factors uniquely through a homomorphism $J(C)\to A$. This universal property converts a curve embedded in an abelian variety into a homomorphism from its Jacobian.

The theta divisor consists of degree-$g-1$ line bundles with a nonzero global section. It is the irreducible image of the symmetric product of a curve $C^{(g-1)}$, with its reduced Cartier divisor structure. The determinant description of the theta divisor supplies equations, multiplicities and pullbacks in families.

For the period matrix of a genus-$g$ compact Riemann surface, there is a vector $\kappa$ depending on the marking and base point such that the Riemann theta function vanishes exactly on this translate of the image of the Abelian sum map. Pairing sides of the cut surface proves that a nonzero pullback $\theta (u(p)-z)$ has a degree-$g$ zero divisor whose Abel sum is $z+\kappa$. At a theta zero that pullback includes the base point. Irreducibility and dimension of the effective-divisor locus give equality of supports; the primitive First Chern class of the theta line bundle makes the divisor reduced.

If $p\mapsto \theta (u(p)-z)$ is not identically zero, its zero divisor $D_z$ has these properties. The logarithmic derivative changes by $-2\pi i{\omega }_j$ across the paired $b_j$ translation, giving zero count $g$. For a variation of $z$, the logarithm changes there by $2\pi i\delta z_j$; integrating by parts in the first-moment contour integral gives $\delta \mathrm{AJ}(D_z)=\delta z$. Thus the difference is constant. Weighted zero divisors, rather than a distinct-zero assumption, make the statement valid at collisions.

The vector of Riemann constants is the parameter-independent class obtained from the zeros of $p\mapsto \theta (u(p)-z)$ using the displayed formula. Under this sign convention the theta divisor is $W_{g-1}-\kappa$. It depends on the base point and normalized symplectic marking; some accounts use its negative. Changing the marking or sign convention must also change the translation formula consistently.

Translate the tangent hyperplane at a smooth point of the theta divisor to the identity of the Jacobian variety. Its annihilator defines the Gauss map. At $L={\mathcal{O}}_X(D)$ with $h^0(L)=1$, the derivative of the Abelian sum map identifies this annihilator with the unique line $H^0(K_X(-D))$. Thus $\Gamma (L)=[\omega ]$ implies $D\le \mathrm{div}(\omega )$. This connects the intrinsic geometry of the theta divisor to the canonical map of the curve.

Normalize the target in the function field of the generically finite Gauss map of a theta divisor; its finite map has an intrinsic branch locus. For a nonhyperelliptic smooth projective curve, this locus is the projective dual variety of the canonical curve. For a hyperelliptic curve of genus at least three, it is the union of the dual hypersurface of its canonical rational normal curve and the $2g+2$ hyperplanes corresponding to branch points. Locally, ramification exchanges two roots of a hyperplane section, or exchanges the two lifts of a root passing through a branch point. Biduality then recovers the curve or its double-cover data.

For a compact Riemann surface of genus $g\ge 1$, the theta divisor in ${\mathrm{Pic}}^{g-1}(X)$ has the displayed multiplicity. A local determinant description of the theta divisor reduces to an $r$ by $r$ matrix vanishing at $L$, where $r=h^0(L)$. Its linear term is cup product with a tangent class in $H^1({\mathcal{O}}_X)$. The invertible cup-product direction for a line bundle on a curve shows that its determinant has order exactly $r$. Consequently the singular locus consists precisely of classes with at least two independent global sections.

For a square evaluation matrix representing a degree-$g-1$ line bundle $L$, remove an invertible minor of size $N-h^0(L)$. Holomorphic row and column operations leave a square matrix $M$ of size $r=h^0(L)$ with zero constant term. Its linear part in a tangent direction $\xi \in H^1({\mathcal{O}}_X)$ is $\xi ⌣-:H^0(L)\to H^1(L)$. The invertible cup-product direction for a line bundle on a curve makes the leading degree-$r$ determinant nonzero, establishing order exactly $r$ rather than just a lower bound.

The tangent cone to a theta divisor at a degree-$g-1$ line bundle $L$ is cut out in $H^1({\mathcal{O}}_X)$ by the determinant of $\xi ⌣-:H^0(L)\to H^1(L)$. Under Serre duality its matrix entries are $⟨\xi ,s_i{t}_j⟩$ for bases of $H^0(L)$ and $H^0(K_X\otimes L^{-1})$. For $h^0(L)=1$, its tangent hyperplane is annihilated by the nonzero holomorphic differential form $s_1{t}_1$.

For $T_q(x)=x+q$, the homomorphism associated to a line bundle on an abelian variety gives ${\lambda }_{\Theta }(q)=T_q^{\ast }\mathcal{O}(\Theta )\otimes \mathcal{O}(\Theta {)}^{-1}$. Restrict to the negative Abel map of an algebraic curve. The restriction of the theta divisor to an Abel curve identifies this restriction with the degree-zero class $q$. Pullback of algebraically trivial line bundles does not change when the embedding is translated. Thus pullback $\rho :\widehat{J(C)}\to J(C)$ satisfies $\rho {\lambda }_{\Theta }=1$. Equal dimensions and properness make ${\lambda }_{\Theta }$ an isomorphism, including on schemes, rather than only on closed points.

For a degree-$g$ divisor class $D$ with $h^0({\mathcal{O}}_C(D))=1$, let $D_0$ be its unique effective representative. Under ${\varphi }_D(P)=[D-P]$, evaluation of the unique global section gives ${\varphi }_D^{\ast }\Theta =D_0$, including multiplicities. If $h^0({\mathcal{O}}_C(D))\ge 2$, every evaluation kernel is nonzero and the entire curve lies in the theta divisor. These restrictions also show that $h^0(\mathcal{O}(\Theta ))=1$: after pulling back to a dominant product of curves, the ratio of any section to the defining section of the theta divisor is constant on each general coordinate curve, hence constant everywhere.

Choose an effective Cartier divisor $E$ of degree at least $g$ on a smooth projective genus-$g$ algebraic curve. For a family $M$ of degree-$g-1$ line bundles, the evaluation map from ${\pi }_{\ast }M(E)$ to ${\pi }_{\ast }(M(E){|}_E)$ is between vector bundles of the same rank. Its determinant vanishes exactly when $H^0(C,M)$ is nonzero. Changing $E$ changes the presentation by cancelling an invertible determinant factor; the zero Cartier divisor is the theta divisor. At a point with $h^0(M)=h^1(M)=1$, the derivative is the cup-product map $H^1({\mathcal{O}}_C)\to \mathrm{Hom}(H^0(M),H^1(M))$. By Serre duality it is nonzero, since the product of nonzero sections of $M$ and $K_C\otimes M^{-1}$ is nonzero. Hence this Cartier divisor is generically reduced.

Suppose a line bundle $L$ of degree $g-1$ on a smooth projective curve has $h^0(L)=h^1(L)=r$. Choose $r$ distinct points where evaluation is invertible for both $H^0(L)$ and $H^0(K_X\otimes L^{-1})$. Such choices exist simultaneously: each determinant defines a nonempty open subset of $X^r$, since independent sections cannot vanish at every point. Let $\xi \in H^1({\mathcal{O}}_X)=H^0(K_X{)}^{\ast }$ be a sum of nonzero weighted evaluation functionals at these points. The Serre duality matrix of $\xi ⌣-$ factors as the product of the two invertible evaluation matrices and an invertible diagonal matrix. Its determinant is nonzero. This gives the nonzero leading determinant needed for the Riemann singularity theorem.

On $C\times C^n$, let ${\Gamma }_i$ be the graph of the $i$th coordinate and $D={\sum }_i{\Gamma }_i$. Successive restriction exact sequences giveWhen $h^0(L)=n$ and $h^1(L)=0$, the determinant of evaluation vanishes at tuples where $H^0(L(-D))$ is nonzero. On distinct points it is the ordinary evaluation determinant; the diagonal factors remove its forced zero when two points coincide. This explains why the formula remains valid at repeated points, using jets instead of repeated evaluations.

Evaluation determinants for $L\otimes P$ and $L$ have a quotient which is a rational section of ${⨂}_i{\mathrm{pr}}_i^{\ast }P$, not intrinsically a scalar. If $P^N\cong {\mathcal{O}}_C$, its $N$th power becomes a rational function after choosing that trivialization. In Cartier divisor notation, write $P={\mathcal{O}}_C(p)$ and $\mathrm{div}(h)=Np$. With raw meromorphic evaluations the scalar expression is $r^N{\prod }_{i}h(z_i)$. The extra product accounts for the line-bundle frames; it is suppressed when evaluation already uses compatible torsion trivializations.

The Abel map of an algebraic curve sends an effective Cartier divisor $D$ to its line bundle ${\mathcal{O}}_C(D)$. Its scheme fibre over a line bundle $L$ is the projective space of one-dimensional subspaces of $H^0(C,L)$. This follows in families: a relative effective Cartier divisor in the fixed class determines a nowhere fibre-zero section of $L$ up to a line bundle from the parameter space, equivalently a line subbundle of $H^0(C,L)\otimes {\mathcal{O}}_T$. Thus the scheme fibre is smooth, not merely reduced on its points. The dimensions can nevertheless jump, so the Abel map of an algebraic curve need not be a smooth morphism.

For an effective divisor on an algebraic curve $D$ on a smooth projective curve, the Zariski tangent space to its symmetric product of a curve is $H^0({\mathcal{O}}_D(D))$. The derivative of the Abelian sum map is the connecting homomorphism of $0\to {\mathcal{O}}_X\to {\mathcal{O}}_X(D)\to {\mathcal{O}}_D(D)\to 0$. Its dual under Serre duality is restriction of holomorphic differential forms to $D$, including all jets at repeated points. Hence its rank is $\mathrm{deg}D+1-h^0({\mathcal{O}}_X(D))$, and its image annihilates $H^0(K_X(-D))$.

At a multiplicity-$r$ point, choose a local parameter and elementary symmetric coordinates $e_1,\dots ,e_r$ for the nearby roots. The tangent principal part is $v={\sum }_{k=1}^r(-1{)}^{k-1}\delta e_k{z}^{-k}$. Newton's identities give $d(\sum z_i^k)=(-1{)}^{k-1}kd{e}_k$. Expanding the primitive of a holomorphic one-form proves the residue formula, including its sign. The dual map is restriction to the length-$r$ scheme, so it sees all jets through order $r-1$.

A smooth projective curve $C\subset A$ generates $A$ by its point differences if and only if it meets every prime Weil divisor. If a prime Weil divisor avoids $C$, its restriction to $C$ has degree zero; constancy of line bundle degree in a family shows that every translate either contains $C$ or misses it. This forces invariance under $C-C$ and puts the generated subgroup inside the line bundle translation stabilizer. If this subgroup were all of $A$, the divisor would be algebraically trivial, contradicting its positive intersection with an ample line bundle. Conversely a proper closed generated subgroup admits a disjoint divisor by the pole divisor avoiding a fiber construction, after translating the fiber to contain $C$.

The dual of an abelian variety represents algebraically trivial line bundles rigidified at the origin. Its group law is tensor product. A group homomorphism $f:A\to B$ induces $\hat{f}:\hat{B}\to \hat{A}$ by pullback of these line bundles.

For an abelian variety and an integer $N$ invertible in the base field, a torsion line bundle $M\in \hat{A}[N]$ has trivial pullback by $[N]$. Choose a trivialization. The descent action of the kernel $A[N]$ on that trivial line bundle acts through a character ${\chi }_M:A[N]\to {\mu }_N$. Define $e_N(P,M)={\chi }_M(P)$. This is independent of rescaling the trivialization and bilinear under translation and tensor product. A trivial character descends the trivial bundle, so $M$ is trivial. Since both groups have order $N^{2\dim A}$, the character map is an isomorphism and the Weil pairing for an abelian variety is perfect. The theta autoduality of a Jacobian turns it into a pairing on two copies of the Jacobian variety's torsion group.

The normalized universal line bundle restricts to the class $\xi$ on $A\times \{\xi \}$ and is trivialized on both zero sections. For a line bundle $L$ on an abelian variety, its homomorphism associated to a line bundle on an abelian variety satisfiesas Picard group classes. A choice of trivialization of $L_0$ normalizes the right side; uniqueness then follows from the Seesaw theorem.

One form of Mumford's rigidity lemma says that if $X$ is complete, $Y$ is connected, and a morphism $f:X\times Y\to Z$ maps one fiber $X\times \{y_0\}$ to a point, then $f$ factors through the projection to $Y$ under the usual pointed hypotheses.

Translation identifies the cotangent sheaf of a group scheme with the constant bundle determined by its cotangent space at the identity:

The Theorem of the Cube says that the third finite difference of a line bundle on an abelian variety is trivial. On $X^3$, it is the alternating tensor product of the pullbacks along the seven nonempty partial-sum maps.

For a line bundle on an elliptic curve over a number field, the geometric cube identity remains exact as a line-bundle identity. The Weil height machine translates it into the displayed bounded-error identity for a chosen logarithmic height, uniformly on the rational triples. This distinction is essential: arbitrary naive heights do not obey an exact quadratic identity. Dividing the identity for $2^{n}P,2^{n}Q,2^{n}R$ by $4^n$ produces an exact identity for the canonical height of an elliptic curve.

If a line bundle on $E^3$ is trivial on each zero-coordinate face, its First Chern class vanishes: the integral second cohomology consists of single-factor second cohomology and cross-products of first cohomology from pairs, and each summand is detected on a coordinate face. The holomorphic exponential sequence then puts the bundle in ${\mathrm{Pic}}^0(E^3)$. The decomposition of first cohomology, including Dolbeault cohomology, identifies this group with ${\mathrm{Pic}}^0(E{)}^3$. Triviality on the three coordinate axes makes all three factors trivial. Applying this cube principle to the alternating partial-sum pullbacks proves the usual addition-map Theorem of the Cube.

For group homomorphisms $f,g:A\to B$, this Picard group class is symmetric and additive in both arguments. The Theorem of the Cube says precisely that the third additive difference of $h\mapsto [h^{\ast }L]$ vanishes, proving additivity of the cross-effect. The Poincaré line bundle expresses it as $(f,{\varphi }_{L}g{)}^{\ast }{P}_B$, so it depends only on the Néron-Severi group class of $L$. Its associated homomorphism is $\hat{f}{\varphi }_{L}g+\hat{g}{\varphi }_{L}f$.

For a line bundle $L$ on an abelian variety and points $x,y$,

For a line bundle $L$ on an abelian variety $X$,is a homomorphism by the Theorem of the square.

For a line bundle $L$ on an abelian variety $A$, this closed subgroup consists of $a$ such that $T_a^{\ast }L\cong L$. It is a stabilizer of a Picard group class, which need not be the same as the stabilizer of an individual Weil divisor representing that class.

For an abelian variety,is the subgroup of translation-invariant line bundles.

The seesaw theorem determines a line bundle on a product from its restrictions to fibers: if it is trivial on all fibers over one factor and on one transverse section, then it is trivial.

A smooth algebraic curve is a one-dimensional algebraic variety whose local ring at every point is regular.

This is the nonsingular subclass of algebraic curves.

For nonzero finite-dimensional spaces $V,W$ of sections of two line bundles on a smooth integral curve over an algebraically closed field, their product span in sections of the tensor product satisfies this bound. At one point choose bases with strictly increasing orders of vanishing. Multiplying every first-space basis vector by the first second-space vector, and then the last first-space vector by the remaining second-space vectors, produces distinct increasing orders and hence independent sections. This proves the dimension inequality needed for the Clifford inequality for curves.

An affine algebraic set is the common zero set $V(I)$ of an ideal in a polynomial ring over a field.

A classical variety is affine if finitely many global regular functions generate the unit ideal and their nonvanishing opens are affine. Localization of global sections on a principal open identifies their coordinate rings with $A_{f_i}$, where $A=\Gamma (X,{\mathcal{O}}_X)$. Choose a finite-type subalgebra of $A$ containing the functions, their unit-ideal coefficients and numerators of generators of every $A_{f_i}$. Its affine variety has the same principal affine charts, whose isomorphisms glue globally. This avoids assuming finite generation of $A$ before affineness is known.