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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]

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.

Given a base point $p_0$, the Abel map takes points of a smooth projective curve to its degree-zero Picard group of a curve. For positive genus it is injective: equality at distinct $p,q$ would give a principal divisor with one simple zero and one simple pole, forcing a degree-one map to the projective line and genus zero. For genus one the Riemann-Roch theorem gives a unique effective degree-one representative of $L(p_0)$ for every degree-zero line bundle $L$, making the map bijective and inducing an abelian group law on the points.

A Cartier divisor is ample when its associated line bundle is ample. The Nakai–Moishezon criterion and Kleiman's criterion characterize this condition numerically.

On a quasi-compact scheme $X$, a line bundle $L$ is ample if each point has an affine neighborhood of the formEquivalently, the affine nonvanishing loci of global sections of positive tensor powers form a basis of a topology. This makes ampleness a condition that global sections recover enough affine local geometry.^[ref]

For a proper scheme over a field, this is equivalent to some positive tensor power being very ample: $L^{\otimes m}$ is the pullback of $\mathcal{O}(1)$ by a closed immersion into projective space. A positive tensor power may be needed. On an elliptic curve, a degree-one line bundle is ample but has only one independent global section, so it cannot itself embed the curve. Its third power has degree three and gives a plane cubic embedding. On an affine scheme, every line bundle is ample under the scheme definition, even when the scheme is not proper.

On a projective scheme, a line bundle is ample exactly when its restriction to every reduced irreducible component is ample. The disjoint union of those components maps finitely and surjectively to the scheme, so finite-surjective descent of the finite pullback of an ample line bundle proves this. The same argument applies to restrictions to any closed subscheme. Thus nilpotent structure does not affect ampleness.

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.

An associated point of a coherent sheaf is a point whose corresponding prime is the annihilator of an element of the module on an affine chart. There are finitely many on a Noetherian scheme. A section of a line bundle avoiding them acts injectively on the sheaf, even when the scheme is nonreduced. Over an infinite field a sufficiently ample linear system supplies such a section.

A morphism between integral varieties is birational if it induces an isomorphism of their function fields, equivalently an isomorphism on suitable dense open subsets. A proper birational morphism to a normal variety satisfies $f_{\ast }{\mathcal{O}}_X={\mathcal{O}}_Y$. For point blowups of smooth surfaces this identifies sections of pulled-back line bundles by the projection formula for sheaves.

Every vector bundle on the projective line over a field splits as a direct sum of line bundles. Choose the largest integer $a$ with $H^0(E(-a))\ne 0$. A corresponding map $\mathcal{O}(a)\to E$ is saturated, since saturation otherwise increases its degree. Inductively split its quotient as ${⨁}_i\mathcal{O}(b_i)$. Twisting the exact sequence by $-a-1$ and using $H^1(\mathcal{O}(-1))=0$ gives $b_i\le a$. Hence all extension groups $H^1(\mathcal{O}(a-b_i))$ vanish, and the sequence splits. The multiset of degrees is determined by the dimensions of all twisted global-section spaces.

For a smooth $n$-dimensional variety $X$ over a field, its canonical line bundle is ${\omega }_X={\bigwedge }^n{\Omega }_{X/k}^1$. It is the bundle of top-degree algebraic differential forms. A nonzero rational section defines a canonical divisor of a smooth variety. This algebraic definition works in arbitrary characteristic and is distinct from restricting the analytic canonical-bundle definition to complex manifolds.

For a real or complex line bundle $L$, the evaluation isomorphism $L^{\ast }\otimes L\cong \mathrm{End}(L)$ identifies the identity endomorphisms with a nowhere-zero global section. Consequently $L^{\ast }\otimes L$ is trivial.

For a vector bundle $E$ of rank $r$ on a smooth algebraic variety, algebraic Chern classes have $c_0(E)=1$, vanish for $i>r$, commute with pullback, and obey the Whitney sum formula for Chern classes for every short exact sequence of vector bundles. A line bundle $\mathcal{O}(D)$ has total class $1+[D]$. The projective-bundle splitting construction makes pullback injective and supplies a filtration by line bundles, so these axioms determine all classes as elementary symmetric functions of their first classes. In particular, $c_i(E^{\ast })=(-1{)}^i{c}_i(E)$ and $c_1(L\otimes M)=c_1(L)+c_1(M)$ in the ordinary Chow ring.

Christian Schnell studies Hodge theory and the geometry and topology of complex algebraic varieties. His Stony Brook research page emphasizes applications of mixed Hodge modules to algebraic geometry.^[ref]

His lecture on the Appell–Humbert theorem describes holomorphic line bundles on a complex torus using a Hermitian form and compatible lattice data. Tensor product becomes addition of the forms and multiplication of the accompanying characters. The classification turns an abstract Picard group into explicit data that also help decide when a torus is projective.^[ref]

If a line bundle on a smooth projective curve has both $h^0(\mathcal{L})>0$ and $h^0({\omega }_C\otimes {\mathcal{L}}^{-1})>0$, the displayed inequality holds. Write these dimensions as $r,s$. The product dimension bound for sections on a curve gives $r+s-1\le g$, and the Riemann-Roch theorem gives $r-s=\mathrm{deg}\mathcal{L}+1-g$. Adding gives $2r\le \mathrm{deg}\mathcal{L}+2$.

Let $C$ be a smooth projective curve over a field $k$ with $H^0(C,{\mathcal{O}}_C)=k$. For a closed point $P$ and a line bundle $L$, the exact sequencehas a quotient of $k$-dimension $[k(P):k]$. Additivity of Euler characteristic therefore gives $\chi (L)-\chi (L(-P))=[k(P):k]$, not necessarily one. Iterating for positive and negative divisor coefficients proveswhere $g={\dim }_k{H}^1(C,{\mathcal{O}}_C)$. Over an algebraically closed field, every closed point has residue degree one and this is the familiar coefficient-sum degree.^[ref]

Serre duality identifies $h^1({\mathcal{O}}_C(D))$ with $h^0({\mathcal{O}}_C(K-D))$, givingThe elementary point modifications supply the Euler-characteristic formula; duality is the additional input turning it into the usual section-counting Riemann-Roch theorem.

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.

In a proper flat family of algebraic curves, if the first sheaf cohomology of a line bundle vanishes on every fibre of an open set, its direct image there is a vector bundle of rank given by the Riemann-Roch theorem and commutes with arbitrary base change. Locally a finite free complex computes both fibre sheaf cohomology groups. Vanishing of the last cokernel makes its last map surjective; splitting that map leaves a finite locally free kernel in degree zero. This is the mechanism behind charts of the Picard scheme of a curve and evaluation descriptions of the theta divisor.

This is the full linear system of divisors, obtained by using every global section of the associated line bundle. For a Cartier divisor on an integral projective variety, the complete linear system $|D|$ consists of its effective representatives under linear equivalence of Cartier divisors. A basis of $H^0(X,{\mathcal{O}}_X(D))$ defines a rational map to projective space.

In characteristic different from two, a local equation for a conic bundle is $z^{\mathsf{T}}Mz=0$ with $M$ symmetric. A fibre is singular exactly when the determinant vanishes. Changing the fibre coordinates multiplies that determinant by a nonzero square, so its zero locus is intrinsically defined even when the quadratic coefficients belong to twisting line bundles.