Line bundle

A line bundle on a scheme is a locally free sheaf of rank one. Its global sections can define a morphism to projective space when they have no common zero.

On a smooth projective curve, a nonzero rational section represents a line bundle by its divisor on an algebraic curve. The degree is the sum of its vanishing orders minus its pole orders, including residue-field degrees over a nonclosed field. Principal divisors on an algebraic curve have degree zero, so this number is independent of the section. A nonzero regular section gives an effective divisor and therefore a nonnegative degree. Degrees add under tensor products and change sign under duality.

A half-density transforms by the square root of the absolute Jacobian under a coordinate change. The product of a half-density with its complex conjugate is an integrable density, so its norm is coordinate-independent without choosing a separate volume form. Half-densities provide an intrinsic configuration-space Hilbert space for a real cotangent polarization.

The orientation line bundle of a smooth $n$-manifold is the real line bundle ${\bigwedge }^{n}TM$. Its transition functions have the signs of the tangent-coordinate transition determinants. A nonzero continuous section specifies an orientation of the manifold. Its orientation double cover records the two possible local orientations.

The total space of the orientation line bundle is orientable even when its base is not: an orientation reversal in a base coordinate transition is matched by a reversal in the line coordinate, so their product has positive sign. Fiberwise multiplication by $1-t$ gives a deformation retraction onto the zero section. For the real projective plane this total space is $(S^2\times \mathbb{R})/((v,t)\sim (-v,-t))$, an oriented noncompact three-manifold homotopy equivalent to the plane.

On $S^2=SU(2)/U(1)$ a degree-$q$ equivariant line bundle corresponds to isotropy character $\mathrm{diag}(e^{it},e^{-it})\mapsto e^{-iqt}$. The Peter-Weyl theorem computes its section multiplicities by the occurrence of weight $-q$ in the SU(2) representation ${\mathrm{Sym}}^n{\mathbb{C}}^2$. Its weights $n,n-2,\dots ,-n$ give the displayed multiplicity-one decomposition. The spinor lines have degrees $-1,+1$ and thus contain all odd $n$, once each.

A line bundle trivial on each open set is determined by unit-valued transition functions on pairwise intersections. Their cocycle identities make the local trivial bundles glue; changing frames changes the cocycle by a coboundary. The tensor product of sheaves multiplies the transition functions, giving the displayed group isomorphism with Čech cohomology.

A line bundle is globally generated if at each point some global section has a germ generating its stalk as a module over the local ring. Equivalently its evaluation map from global sections tensored with the structure sheaf is surjective.

The displayed line bundle is globally generated, but no finite family of global sections generates it. A family of $N$ sections restricts to $N$ linear forms on ${\mathbb{P}}^m$; for $m\ge N$, these have a common zero. This shows why finite global generation on a quasi-compact scheme requires its finiteness hypothesis.

A globally generated line bundle on a quasi-compact scheme is generated by finitely many global sections. Each point has a section generating on an open neighbourhood, and quasi-compactness gives a finite subcover. This supplies a map into a fixed finite-dimensional projective space associated with that line bundle.

The tensor product of two globally generated line bundles is globally generated. At a point choose generator germs arising from a global section of each factor. Their tensor is the germ of a global tensor-product section and generates the rank-one tensor-product stalk. Iteration gives the same property for every positive tensor power, on an arbitrary scheme.

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.

The twisting sheaf $\mathcal{O}(d)$ on $\mathrm{Proj}k[x_0,\dots ,x_n]$ is associated with the shifted graded module $k[x_0,\dots ,x_n](d)$. For $d\ge 0$, its global sections are the homogeneous polynomials of degree $d$.

On projective space the twisting sheaf $\mathcal{O}(d)$ is the $d$th tensor power of $\mathcal{O}(1)$ for $d\ge 0$, and the corresponding dual power for $d<0$. Twisting another sheaf means tensoring it with this line bundle.

The local frame $X_i^m$ on $X_i\ne 0$ identifies a section with a homogeneous rational function of degree $m$ on the punctured affine cone. Homogenizing a local rational expression and cancelling common factors writes every nonzero section as a quotient of coprime homogeneous polynomials $F/G$ with $\mathrm{deg}F-\mathrm{deg}G=m$. Conversely, such a rational function regular on the cone gives local coefficients $r/X_i^m$, which glue with the twisting sheaf's transition functions of a vector bundle.

For $n\ge 1$, the top Čech cohomology of the standard affine cover of projective space is the degree-$m$ quotient of the Laurent ring by the sum of rings in which at least one variable has not been inverted. Its basis consists of Laurent monomials $X_0^{a_0}\cdots X_n^{a_n}$ with every $a_i<0$ and ${\sum }_i{a}_i=m$. Hence it vanishes for $m\ge -n$ and has dimension $\left(\genfrac{}{}{0ex}{}{-m-1}{n}\right)$ for $m\le -n-1$.

Over any field, the canonical bundle of projective space is $\mathcal{O}(-n-1)$. Indeed, dualizing the Euler sequence gives $0\to {\Omega }_{{\mathbf{P}}^n}^1\to \mathcal{O}(-1{)}^{\oplus (n+1)}\to \mathcal{O}\to 0$; taking determinants gives the result. Hence Serre duality identifies $H^i({\mathbf{P}}^n,\mathcal{O}(m))$ with the dual of $H^{n-i}({\mathbf{P}}^n,\mathcal{O}(-m-n-1))$.

On the standard charts of projective space, wedge the coordinate differentials in increasing order. The change $X_0/X_i=(X_i/X_0{)}^{-1}$ and $X_j/X_i=(X_j/X_0)/(X_i/X_0)$ gives the displayed Jacobian determinant. Multiplying the $i$th local generator by $(-1{)}^i$ removes this sign and yields the transition functions of the twisting sheaf on projective space with degree $-n-1$.

For a hyperplane $H\subset {\mathbf{P}}^n$ with inclusion $i$, multiplication by its defining linear homogeneous polynomial gives the short exact sequence of sheavesThe long exact sequence in sheaf cohomology relates twists in consecutive degrees and consecutive dimensions. Surjectivity of restriction on global homogeneous polynomials, together with eventual positive-twist vanishing, proves vanishing of intermediate cohomology by descending induction. The top-degree dimensions satisfy $h_n(m-1)=h_n(m)+h_{n-1}(m)$ for $n\ge 2$, where $h_r(m)=\dim H^r({\mathbf{P}}^r,\mathcal{O}(m))$.

The hyperplane exact sequence for twisting sheaves identifies consecutive top-degree groups whenever the hyperplane's preceding-degree group vanishes. Induction on dimension, starting with the surjective restriction of constants to a point on ${\mathbf{P}}^1$, propagates the degree-zero top vanishing down to $m=-n$. Its top-degree surjections also propagate vanishing to every positive twist. This proof does not apply the negative-twist global-section formula to ${\mathbf{P}}^0$.

The Picard group is the group of isomorphism classes of line bundles under tensor product.

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 curve, the degree-zero Picard group consists of line bundles of degree zero up to isomorphism, with tensor product as addition. A nonzero rational section of a line bundle identifies each line bundle with $\mathcal{O}(D)$ for a divisor on an algebraic curve; degree is the sum of multiplicities weighted by residue-field degrees. Over an algebraically closed field, every degree-zero divisor is a sum of differences of points.

Let $C$ be the smooth projective model over ${\mathbb{F}}_2$ of $y^2+y=x^5+x+1$. Its only rational point is its unique point $p_{\infty }$ at infinity. At that point $x,y$ have pole orders two and five. The degree-two closed point $D$ defined by $y=0$, $x^2+x+1=0$ gives a nonzero class $[D-2p_{\infty }]$. Indeed a function with poles bounded by $2p_{\infty }$ has the form $a+bx$: in the affine coordinate ring every function is uniquely $A(x)+B(x)y$, and the two possible pole orders have opposite parity. No nonzero $a+bx$ vanishes at both roots of $x^2+x+1$. Thus the Abel map of a pointed smooth projective curve on rational points has trivial image here, but the degree-zero Picard group of a curve is nontrivial. Generation by point differences needs an algebraically closed field, or closed points with their residue degrees instead of just rational points.

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.

For a smooth projective geometrically connected algebraic curve $C/k$, this functor assigns to a scheme $T$ the line bundles on $C\times T$ having degree $d$ on each geometric fibre, modulo bundles pulled back from $T$. One generally takes the associated sheaf for faithfully flat covers. A rational point of $C$ permits normalization of every line bundle along that point; normalized bundles have no automorphisms preserving their normalization and satisfy effective descent. The resulting functor is represented by the degree-$d$ component of the Picard scheme of a curve.

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.

A degree-$g$ line bundle with vanishing first sheaf cohomology has exactly one section up to scale, by Riemann-Roch theorem. Its zero divisor is unique and varies in families by cohomology and base change for line bundles on a curve. For a fixed divisor $B$ of degree $N-g$, the open subset of $C^{(g)}$ where $\mathcal{O}(E)$ has vanishing first cohomology gives a chart in degree $N$, by $E\mapsto \mathcal{O}(B+E)$. Different charts are glued by the unique residual divisor of $\mathcal{O}(B+E-B^{\prime })$ wherever that line bundle is again nonspecial. Normalizing the universal bundles along a fixed rational point gives their canonical compatible gluing and a representing scheme for the Picard functor of a curve.

The Picard group of a commutative ring $A$ consists of isomorphism classes of finitely generated projective modules that have rank one at every prime ideal. Tensor product is the group law, $[A]$ is the identity, and the module dual supplies the inverse. For an integral domain, every such module can be represented by an invertible fractional ideal after choosing a basis over the fraction field.

The holomorphic Picard group consists of isomorphism classes of holomorphic line bundles, with tensor product as its group law. The trivial bundle is the identity and dualization gives inverses. Transition functions identify it with $H^1(X,{\mathcal{O}}_X^{\ast })$. On the complex projective line every class is $\mathcal{O}(n)$, so degree gives an isomorphism with $\mathbb{Z}$.

For a rank-two holomorphic vector bundle on a nonempty complex manifold, this map from ${\mathrm{Pic}}_{\mathrm{hol}}(X)\times \mathbb{Z}$ to ${\mathrm{Pic}}_{\mathrm{hol}}(\mathbb{P}(E))$ is injective. Fibre degree detects $n$. If $p^{\ast }M$ is trivial, a nonvanishing holomorphic section is constant along every compact projective fibre in a local bundle chart, so it descends to a nonvanishing section of $M$. The proof requires no global section of $p$.

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.

On an irreducible variety, a Cartier divisor is a section of ${\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$: its local rational equations differ by regular units. Their ratios define an invertible sheaf, and a global rational equation gives the trivial class. Conversely a rational trivialization of an invertible sheaf gives local equations. The rational-function sheaf is flasque, so its long exact sequence in sheaf cohomology also identifies the quotient with $H^1(X,{\mathcal{O}}_X^{\ast })$.

The Néron-Severi group is the quotient $\mathrm{Pic}(X)/{\mathrm{Pic}}^0(X)$ of the Picard group by line bundles algebraically equivalent to zero.

A line bundle is very ample when its global sections define a closed embedding into projective space.

Over a field, a line bundle is very ample when it is the pullback of $\mathcal{O}(1)$ by a closed immersion into projective space. It is an ample line bundle, but an ample bundle need not itself be very ample.

A line bundle $L$ is ample when some positive tensor power $L^{\otimes m}$ is very ample.

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 finite morphism $f:X\to Y$, the pullback of an ample line bundle is ample. If $f$ is also surjective, ampleness is equivalent in both directions. These results hold for proper schemes over a field, including nonreduced schemes. Cohomological ampleness and the exact finite pushforward prove the forward direction; coherent-sheaf devissage gives finite-surjective descent. In particular, a finite map to projective space makes the pulled-back hyperplane bundle ample.

If $A$ is an ample line bundle on a projective variety and $L$ is any line bundle, then $A^{\otimes m}\otimes L$ is very ample for all sufficiently large integers $m$. Taking $L$ to be trivial also makes $A^{\otimes m}$ very ample.

A basepoint-free vector space of global sections $V\subseteq H^0(X,L)$ defines the Kodaira map $X\to \mathbb{P}(V^{\ast })$ by evaluating the sections at each point.