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For a smooth integral variety, every Weil divisor is a Cartier divisor, so the Picard group equals the divisor class group. If $U$ is obtained by removing irreducible Weil divisors $D_j$, the localization sequence for the divisor class group gives the displayed exact sequence. A rational trivialization of a line bundle on $U$ has divisor supported on the removed divisors, describing the kernel concretely.

Over an algebraically closed field, remove a smooth hyperplane conic $C$ from a smooth quadric surface $Q$. The rulings of a smooth quadric surface generate $\mathrm{Pic}(Q)={\mathbb{Z}}^2$, and $C$ has class $(1,1)$. Picard-group localization on a smooth variety gives the quotient by $\mathbb{Z}(1,1)$. The restricted ruling bundle ${\mathcal{O}}_Q(1,0)$ is a generator, exhibiting a nontrivial line bundle on an affine surface.

For any field $k$, degree identifies the Picard group of the projective line with $\mathbb{Z}$, generated by $\mathcal{O}(1)$. A closed point on the affine chart is cut out by an irreducible polynomial $f(t)$, whose divisor is that point minus $\mathrm{deg}(f)$ times infinity. Hence every divisor is linearly equivalent to its degree times infinity. Principal divisors have degree zero, so the degree map is injective as well as surjective. Thus a line bundle of degree $d$ is $\mathcal{O}(d)$.

For a smooth projective algebraic curve of genus $g$, this smooth proper $g$-dimensional scheme represents degree-$d$ line bundles in families. Charts come from nonspecial degree-$g$ bundles: their first sheaf cohomology vanishes, so the Riemann-Roch theorem gives exactly one independent global section, whose zero Cartier divisor varies algebraically. Twisting by fixed line bundles gives charts covering every degree-$d$ class. For degree zero the tensor product gives the Jacobian variety; other components are its torsors.

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.

Let $q(\varphi )=\mathrm{deg}\varphi$ for nonzero elements of the homomorphism group of elliptic curves, and $q(0)=0$. For an elliptic curve $E$, the degree-two coordinate $x$ on a Weierstrass model has a double pole at its identity and identifies a generic point only with its inverse. Thus on $E\times E$,where the two diagonals are $P=Q$ and $P=-Q$. The resulting line bundle isomorphism is $a^{\ast }{\mathcal{O}}_E(O)\otimes s^{\ast }{\mathcal{O}}_E(O)\cong {\mathrm{pr}}_1^{\ast }{\mathcal{O}}_E(O{)}^{\otimes 2}\otimes {\mathrm{pr}}_2^{\ast }{\mathcal{O}}_E(O{)}^{\otimes 2}$, for the sum and difference maps $a,s$. Pull it back along $(\varphi ,\psi )$ and take degrees. This gives the degree parallelogram law, even when one homomorphism equals plus or minus the other: bundle restriction remains valid in these cases.

It implies $q(n\varphi )=n^{2}q(\varphi )$ by the second-difference recursion, and polarization gives the symmetric bilinear form $B(\varphi ,\psi )=(q(\varphi +\psi )-q(\varphi -\psi ))/4$. To check additivity, the parallelogram law gives $B(x+z,y)+B(x-z,y)=2B(x,y)$; interchange $x,z$ and use oddness to get $B(x+z,y)-B(x-z,y)=2B(z,y)$. Adding proves bilinearity. Since a nonzero isogeny has positive degree, $q$ is positive on every nonzero element. On every finite-dimensional rational span the matrix of $B$ is rational. Positivity on rational vectors implies nonnegativity on real vectors by density; a nontrivial kernel of a rational matrix would contain a nonzero rational vector. Hence the real extension is positive definite. This proves the positive-definite quadratic form property, rather than only a parallelogram identity.

Prequantization realizes a symplectic manifold using a Hermitian line bundle with connection whose curvature is $i\omega /\hslash$ in the convention ${\iota }_{X_f}\omega =df$. Its existence requires integral symplectic periods. On sections, the Kostant-Souriau prequantum operator represents every classical observable and its Poisson bracket, but the resulting states depend on the full phase space. A polarization in geometric quantization removes this excess of variables.

A prequantum line bundle is a Hermitian line bundle over a symplectic manifold with unitary connection on a vector bundle of the displayed curvature. With $\omega =-d\theta$, a local connection is $\nabla =d-i\theta /\hslash$. Reversing the Hamiltonian-vector-field convention or passing to the dual line bundle reverses the corresponding curvature convention. Integrality of $\omega /(2\pi \hslash )$ on closed two-cycles is the topological existence condition.

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.

The projective bundle of a real or complex vector bundle $E\to X$ has fiber the space of one-dimensional subspaces of $E_x$. It carries a tautological bundle $\lambda \subset p^{\ast }E$; in the real case this is the projectivization of a real vector bundle.

The same construction applies to an algebraic vector bundle over an algebraic variety: local trivializations glue products with projective space. In the lines convention its universal tautological bundle is a line bundle; the alternative projectivization by quotients convention parametrizes one-dimensional quotient spaces. Both conventions give dimension $\dim X+\mathrm{rank}(E)-1$ and a proper morphism to the base.

With the quotient convention, projective space represents the functor of surjections from a rank-$n+1$ free sheaf to a line bundle, modulo isomorphisms of the target compatible with the map. On an affine base the target may be a nontrivial rank-one projective module, so a tuple of elements generating the unit ideal is only the description when that target is trivial. Where the image of the $i$th basis vector generates the target, divide the other images by it to get the usual affine coordinates. Infinitesimal variations of the quotient form $\mathcal{H}om(\ker q,\mathcal{L})$ and give the Euler sequence.

This convention for a projective bundle parametrizes one-dimensional quotients of the fibres of a vector bundle $E$. Its universal quotient is ${\pi }^{\ast }E\twoheadrightarrow {\mathcal{O}}_{\mathbb{P}(E)}(1)$, and ${\pi }_{\ast }\mathcal{O}(1)=E$. Tensoring $E$ by a line bundle on the base leaves the projective bundle unchanged but tensors its universal quotient by the pullback of that line bundle. The alternative convention of holomorphic projectivization by lines replaces $E$ by its dual and reverses the tautological sign.

For a morphism of varieties $f:X\to Y$, pull back local defining rational functions of a Cartier divisor $D$ whenever they remain nonzero on each component of $X$. Their ratios remain regular units, defining $f^{\ast }D$. Equivalently, pull back the associated line bundle and its rational section. A dominant morphism always permits this construction.

On an integral scheme with generic point $\eta$ and function field $K$, a rational section of a line bundle $\mathcal{L}$ is an element of the one-dimensional $K$-space ${\mathcal{L}}_{\eta }$. A nonzero rational section writes as $a_i{e}_i$ in local trivializations, with $a_i\in K^{\ast }$. Ratios of these coefficients are regular units, and therefore define a Cartier divisor.

A relative effective divisor of degree $m$ on a smooth projective curve is an effective Cartier divisor in $C\times T$ that is finite and flat of degree $m$ over $T$. Its associated line bundle has degree $m$ on every fibre. The symmetric product of a curve $C^{(m)}$ represents these families and carries their universal divisor. Locally on a smooth coordinate chart, the divisor is described by a monic polynomial, whose coefficients are elementary symmetric functions of its points; this description includes collisions and works in positive characteristic.

Dualize the relative tautological line bundle to obtain the positive relative hyperplane line bundle. It restricts to $\mathcal{O}(1)$ on each projective fibre. This sign depends on the convention that projectivization parametrizes lines; fixing the convention avoids confusing it with the tautological line itself.

For a closed point $p$ on a smooth projective curve, the quotient in $0\to L(-p)\to L\to L{|}_p\to 0$ has $k$-dimension $[k(p):k]$. Additivity of Euler characteristic therefore changes $\chi (L)$ by that residue-field degree. Every line bundle is associated with a divisor, so iteration proves the displayed formula. The dualizing sheaf on a smooth projective curve identifies $h^1(L)$ with $h^0({\Omega }_C^1\otimes L^{-1})$ and converts this Euler-characteristic formula into the usual divisor form of Riemann-Roch theorem.