Algebraic group

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 over a field is a complete, geometrically connected group variety; equivalently it is a connected projective algebraic group in the usual smooth variety setting. Its group operation is necessarily commutative, a consequence of rigidity rather than an extra condition hidden in the name. An elliptic curve with a chosen origin is a one-dimensional abelian variety; products of elliptic curves give higher-dimensional examples. The additive and multiplicative algebraic groups are not complete and hence are not abelian varieties. Their geometry links line bundles, torsion elements and the Jacobian variety of a curve to questions in number theory.^[ref]

Over an algebraically closed field $k$, an ample line bundle $L$ on an abelian variety $A$ defines a homomorphismwhere $A^{\vee }$ is the dual abelian variety and $t_x$ is translation by $x$. This isogeny is a polarization of an abelian variety. Over a general field, a polarization is an isogeny that has this form after extending to an algebraic closure; an inducing ample line bundle need not descend to the original field.^[ref]

A principal polarization is an isomorphism $A\to A^{\vee }$. An elliptic curve with its degree-one origin divisor gives a basic example. A polarization supplies positivity data relating the geometry of line bundles to the algebra of isogenies; twisting $L$ by a degree-zero line bundle does not change ${\lambda }_L$.

A principal polarization of an abelian variety $A$ is a polarization of an abelian variety $\lambda :A\to A^{\vee }$ that is an isomorphism. Equivalently, its isogeny has degree one. It identifies $A$ with its dual abelian variety through positivity supplied by an ample line bundle; an arbitrary isomorphism with the dual need not be a polarization of an abelian variety.^[ref]

For an elliptic curve $E$ with origin $0$, the degree-one line bundle ${\mathcal{O}}_E(0)$ gives a principal polarization. Replacing it by ${\mathcal{O}}_E(n\cdot 0)$, for positive $n$, multiplies the corresponding map to the dual by $n$ and gives degree $n^2$. Thus positivity alone does not make a polarization of an abelian variety principal: the size of its kernel matters.

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.

A local form of the Mumford rigidity lemma says: if $X$ is a complete integral variety over an algebraically closed field, and a morphism $f:X\times Y\to Z$ sends one fibre $X\times \{y_0\}$ to a point, then on $X\times U$ for some open neighbourhood $U$ of $y_0$, it factors through the projection to $U$. To see the mechanism, choose an affine neighbourhood of that image point. Properness of the projection supplies $U$ whose whole product maps into that neighbourhood. Regular functions on the complete integral $X$ are constant, forcing the factorization there. Pointed global versions add further hypotheses; the local conclusion alone should not be silently replaced by a global one.^[ref]

An invariant differential on a group scheme is a differential one-form preserved by every left translation. For a smooth group scheme $G$ over a field $k$, translation identifiesThus a covector at the identity extends uniquely to a left-invariant form. On the additive algebraic group the invariant form is $dx$; on the multiplicative algebraic group it is $dx/x$, since $d(ax)/(ax)=dx/x$. An abelian variety of dimension $g$ therefore has $g$ linearly independent invariant one-forms, providing the local linear data behind its global group geometry.

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 $A$ and translations $T_x(a)=a+x$, the Theorem of the square gives an isomorphism of line bundlesConsequently $x\mapsto T_x^{\ast }L\otimes L^{-1}$ is a group homomorphism from $A$ to its dual abelian variety. These are isomorphism classes: no canonical trivialization is supplied without normalization data. On an elliptic curve, translating a degree-one divisor changes its degree-zero difference class additively, illustrating the identity. The Theorem of the Cube controls the additional compatibility when the translation points themselves vary.^[ref]

For a line bundle $L$ on an abelian variety $A$, setThe translated difference is algebraically trivial, so its value belongs to the dual abelian variety $\hat{A}={\mathrm{Pic}}^0(A)$ rather than merely the whole Picard group. The Theorem of the square makes this a group homomorphism. Tensoring line bundles adds their homomorphisms; an algebraically trivial bundle gives the zero map. An ample line bundle gives a finite surjective map, a polarization of an abelian variety; its kernel measures translations preserving the bundle class.

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 $A$, the Identity component of the Picard group ${\mathrm{Pic}}^0(A)$ consists of algebraically trivial line bundles and is represented by the dual abelian variety. In terms of translations, it isThus these bundles are translation-invariant as isomorphism classes. This does not say their individual global sections are invariant, nor that every such bundle is trivial. On an elliptic curve, $\mathcal{O}(p-q)$ is a degree-zero bundle; it is trivial exactly when its class $p-q$ is the identity.

A useful form of the Seesaw theorem determines a line bundle on a product from its fibre restrictions. Let $X$ be a complete geometrically integral variety with a base point $x_0$, and let $L$ be a line bundle on $X\times Y$ trivial on each geometric fibre $X\times \{y\}$. For a reduced variety $Y$, constancy of the fibrewise section dimension and cohomology and base change imply that $L$ is pulled back from $Y$; restricting to $\{x_0\}\times Y$ identifies the bundle being pulled back. If that transverse restriction is also trivial, then $L$ is trivial. This is useful for verifying normalized universal bundle identities by checking fibres; matching only one fibre in each direction is insufficient.