The sheaf $\OO_X(D)$ and effective divisors as sections (Tag D-DIVOD)
For \(D\) a divisor on \(X\), the sheaf \(\OO_X(D)\) is the subsheaf of the constant sheaf \(\mck\) given on an open \(U\) by \[ \OO_X(D)(U) = \ts{ f \in \unitsof{\mck(U)} \st \div(f) + D \geq 0 } \union \ts{0} . \] Written from Cartier data \(\ts{(U_i, f_i)}\) it is the \(\OO_X\)-submodule of \(\mck\) generated by \(\inverseof{f_i}\) on \(U_i\), hence invertible.
(Divisor of a section and the Riemann--Roch space)
A nonzero global section \(s\) of \(\OO_X(D)\) is a rational function \(f\) with \(\div(f) + D \geq 0\), and its divisor is the effective divisor \(\div(s) = \div(f) + D\), linearly equivalent to \(D\). For a line bundle \(\mcl\) with a nonzero rational section \(s\), \(\div(s)\) is defined by the orders of \(s\) in local trivializations, and \(\mcl \cong \OO_X(\div(s))\). The Riemann–Roch space of \(D\) is \(L(D) = H^0(X, \OO_X(D))\), and the complete linear system associated to a section \(s \neq 0\) of \(\mcl\) is \(\abs{\div(s)} = \PP H^0(X, \mcl)\).
\(\OO_X(D_1 + D_2) \cong \OO_X(D_1) \tensor \OO_X(D_2)\) and \(\OO_X(-D) \cong \dualof{\OO_X(D)}\), and \(\OO_X(D_1) \cong \OO_X(D_2)\) exactly when \(D_1 \sim D_2\). A divisor \(D\) is effective if and only if \(\OO_X(D)\) has a global section \(s\) with \(\div(s) = D\). For \(D\) effective the subscheme it cuts out has ideal sheaf \(\mci_D \cong \OO_X(-D)\), giving \[ 0 \to \OO_X(-D) \to \OO_X \to \OO_D \to 0 . \]
A nonzero section of \(\OO(D)\) is a rational function \(f\) with \(\div(f)\ge-D\). The map \(f\mapsto\div(f)+D\) is a bijection from nonzero global sections of \(\OO(D)\) modulo nonzero scalars to effective divisors linearly equivalent to \(D\), which gives \(\abs{D} = \PP H^0(X, \OO(D))\).
For a hypersurface \(D\subseteq\PP^n\) of degree \(d\), twisting the exact sequence by \(\OO(m)\) gives \(\chi(\OO_D(m))=\binom{m+n}{n}-\binom{m-d+n}{n}\), the Hilbert polynomial of \(D\). On a nonsingular projective curve \(X\) over an algebraically closed field and a closed point \(p\), the sequence \(0\to\OO_X(D-p)\to\OO_X(D)\to k(p)\to0\) gives \(\chi(\OO_X(D))=\chi(\OO_X(D-p))+1\), hence \(\chi(\OO_X(D))=\deg D+\chi(\OO_X)\).
Cartier divisors, invertible sheaves, and the Picard group (Tag PR-DIVLB)
\(D \mapsto \OO_X(D)\) is an injection \(\CaCl(X) \injects \Pic(X)\), and it is a bijection onto the invertible subsheaves of \(\mck\) [@Har10a, Proposition II.6.13]. When \(X\) is integral every invertible sheaf is such a subsheaf, so \(\CaCl(X) \cong \Pic(X)\) [@Har10a, Proposition II.6.15].
\(\Pic(X) \cong H^1(X, \unitsof{\OO_X})\) (P-AGH345PICH1), and the sequence \[ 0 \to \unitsof{\OO_X} \to \unitsof{\mck} \to \unitsof{\mck}/\unitsof{\OO_X} \to 0 \] has cohomology sequence \(\unitsof{\mck(X)} \to \Div_{\mathrm{Ca}}(X) \to \Pic(X) \to H^1(X, \unitsof{\mck})\).
(Divisors, line bundles with sections, and class groups)
Let \(X\) be an integral separated Noetherian scheme with function field \(K\), and write \(\mathcal{P}(X)\) for the group of isomorphism classes of pairs \((\mathcal{L}, s)\) with \(\mathcal{L}\) invertible and \(s\) a nonzero rational section of \(\mathcal{L}\), under tensor product.
\(\Div_{\mathrm{Ca}}(X) \to \mathcal{P}(X)\), \(D = \{(U_i, f_i)\} \mapsto (\OO_X(D), 1)\), is an isomorphism, with inverse \((\mathcal{L}, s) \mapsto \{(U_i, \phi_i(s)^{-1})\}\) for local trivializations \(\phi_i\). For an effective \(D\) with invertible ideal sheaf \(\mci_D\), \(\OO_X(D) = \mci_D^{\vee}\) and \(1\) is the canonical section vanishing on \(D\).
If \(X\) is regular in codimension one, \(\Div_{\mathrm{Ca}}(X) \to \Div(X)\), \(\{(U_i, f_i)\} \mapsto \sum_Y v_Y(f_i) Y\) with \(U_i \cap Y \neq \emptyset\), is a homomorphism, and \((\mathcal{L}, s) \mapsto \div s\) is the composite with the inverse of part 1.
Under these maps, \(t \in \unitsof{K}\) goes to the principal Cartier divisor \(\{(X, t)\}\), the pair \((\OO_X, t)\), and the principal Weil divisor \(\div t\), so they descend to \(\CaCl(X) \to \Cl(X)\) and \(\CaCl(X) \cong \Pic(X)\).
If every local ring \(\OO_{X,x}\) is a unique factorization domain, then \(\CaCl(X) \cong \Cl(X) \cong \Pic(X)\), with inverse \(D \mapsto (\OO_X(D), 1)\) [@Har10a, Proposition II.6.11].
Both halves are the same observation: gluing data for a line bundle is a cocycle \(g_{ij} \in \unitsof{\OO}(U_{ij})\), and Cartier data \(f_i\) produces the cocycle \(g_{ij} = f_i/f_j\). Saying “a Cartier divisor is a line bundle together with a choice of rational section” is the cleanest one-line answer, and it explains the failure of surjectivity for non-integral \(X\): a line bundle with no rational section is not a divisor.
Combined with the injection \(\CaCl \injects \Cl\), the chain to recite is \[ \Pic(X) \cong \CaCl(X) \injects \Cl(X) , \] an isomorphism throughout when \(X\) is locally factorial. The gap at the second arrow is measured by the singularities.
A Cartier divisor \(D\) gives the invertible sheaf \(\OO(D)\) with the rational section \(1\), and an invertible sheaf \(\mcl\) with a nonzero global section \(s\) gives the effective divisor \(\operatorname{div}(s)\), its zero locus. Linear equivalence of divisors corresponds to isomorphism of invertible sheaves, because changing the section by a rational function \(f\) changes the divisor by \(\operatorname{div}f\).
From sections to a map
Linear systems, base loci, and base-point freeness (Tag D-DIVLINSYS)
The complete linear system of \(D\) is the set of effective divisors linearly equivalent to it, \[ \abs{D} = \ts{ D' \geq 0 \st D' \sim D } \cong \PP H^0(X, \OO_X(D)) . \] A linear system is a linear subspace of \(\abs{D}\), that is, \(\PP V\) for a subspace \(V \subseteq H^0(X, \OO_X(D))\). A point \(p\) is a base point when \(p \in \supp D'\) for every \(D'\) in the system; the base locus \(\Bs\abs{D}\) is the set of base points, and the system is base-point free when this is empty.
(Sections and degree on a curve)
Let \(D\) be a divisor on a nonsingular projective curve \(X\) over \(k = \bar k\). If \(\abs{D} \neq \emptyset\), that is \(\ell(D) \neq 0\), then \(\deg D \geq 0\). If moreover \(\deg D = 0\), then \(D \sim 0\) and \(\mcl(D) \cong \OO_X\). [@Har10a]
(Degenerate)
A morphism \(X \to \PP^n\), and equivalently the invertible sheaf with sections or the linear system \(\abs{V}\) defining it, is degenerate if the image of \(X\) is contained in a hyperplane.
\(p\) is a base point of \(\PP V\) exactly when \(s_p \in \mfm_p \mcl_p\) for every \(s \in V\). So \(\abs{D}\) is base-point free if and only if \(\OO_X(D)\) is generated by global sections.
The divisors in \(\abs{D}\) are the zero loci of the nonzero sections of \(\OO_X(D)\), two sections giving the same divisor if and only if they differ by a nonzero scalar; so \(\dim\abs{D}=h^0(\OO_X(D)) - 1\), which on a curve Riemann–Roch computes as \(\deg D-g+h^0(\OO_X(K-D))\).
The morphism to projective space attached to a linear system (Tag T-DIVMAPPN)
Let \(X\) be a scheme over a ring \(A\), \(\mcl\) an invertible sheaf on \(X\), and \(s_0, \dots, s_n \in H^0(X, \mcl)\) sections that generate \(\mcl\). Then there is a unique \(A\)-morphism \[ \varphi : X \to \PP^n_A, \qquad p \mapsto [s_0(p) : \cdots : s_n(p)] , \] with \(\mcl \cong \varphi^* \OO_{\PP^n}(1)\) and \(s_i = \varphi^*(x_i)\) under this isomorphism. Conversely, every \(A\)-morphism \(\varphi : X \to \PP^n_A\) arises this way, from \(\mcl = \varphi^*\OO(1)\) and \(s_i = \varphi^*(x_i)\). [@Har10a, Theorem II.7.1]
(Linear systems with base points)
Let \(X\) be a \(k\)-scheme, \(V \subseteq H^0(X, \mcl)\) a subspace with basis \(s_0, \dots, s_n\), and \(B = \bigcap_i V(s_i)\) its base locus. The sections restricted to the open set \(X \sm B\) generate \(\mcl\) there, so they define a morphism \(\varphi_V : X \sm B \to \PP^n_k\), which is \(\PP V\) up to the choice of basis. On the open set \(D(s_i) \subseteq X \sm B\) where \(s_i\) does not vanish, \(\varphi_V\) maps into the standard chart \(D(x_i)\) and is given by \(x_j/x_i \mapsto s_j/s_i\).
(Degrees on curves)
Let \(X\) be a nonsingular projective curve over \(k = \bar k\) and \(V \subseteq H^0(X, \mcl)\) base-point free.
If \(\dim V = 2\), the morphism \(\varphi_V : X \to \PP^1\) is finite of degree \(\deg \mcl\).
If \(\mcl\) is very ample and \(V = H^0(X, \mcl)\), then \(\varphi_V\) embeds \(X\) as a curve of degree \(\deg \mcl\) in \(\PP^n\), and \(\mcl \cong \varphi_V^* \OO(1)\).
[@Har10a, Proposition II.6.9, Theorem II.7.1]
Erratum: the source states the degree of the pencil map without assuming \(V\) base-point free. If a two-dimensional \(V\) has base points, the fixed part \(F\) of the pencil is an effective divisor of positive degree, and the morphism \(X \to \PP^1\) that \(V\) induces has degree \(\deg \mcl - \deg F\).
(Closed immersion criterion)
Let \(X\) be a projective scheme over \(k = \bar k\) and \(V \subseteq H^0(X, \mcl)\) base-point free. \(\varphi_V\) is a closed immersion exactly when the system separates points and separates tangent vectors:
for all closed points \(p \neq q\) there is \(D \in \abs{V}\) with \(p \in \supp D\) and \(q \notin \supp D\);
for every closed point \(p\) and every \(0 \neq t \in T_p X\) there is \(D \in \abs{V}\) with \(p \in \supp D\) and \(t \notin T_p(D)\), where \(D\) is regarded as a closed subscheme.
[@Har10a, Proposition II.7.3]
Base-point freeness is exactly what makes the formula define a point of \(\PP^n\) at all: some coordinate must be nonzero. The two separation conditions are then injectivity and injectivity on tangent spaces, which is what a closed immersion is.
The choice of \(V\) matters, not just \(\mcl\): taking the complete system gives the largest target, and a proper subspace gives a linear projection of that image. Standard examples are \(\abs{\OO_{\PP^1}(d)}\), which embeds \(\PP^1\) as the rational normal curve in \(\PP^d\), and \(\abs{\OO_{\PP^n}(2)}\), which gives the Veronese embedding. On a curve the two separation conditions become the numerical statement about \(\ell(D - p - q)\).
For a basis \(s_0,\ldots,s_n\) of a linear system \(V\subseteq H^0(X,\mcl)\), the morphism sends \(x\) to \([s_0(x):\cdots:s_n(x)]\), with the values taken in a trivialization of \(\mcl\) near \(x\). It is defined exactly off the base locus of \(V\); for \(X\) projective over an algebraically closed field, it is a closed immersion exactly when \(V\) separates points and tangent vectors.
On a smooth projective curve \(C\) of genus \(g\), \(\abs{D}\) is base-point free exactly when \(\ell(D-p)=\ell(D)-1\) for every point \(p\), and very ample exactly when \(\ell(D-p-q)=\ell(D)-2\) for all points \(p,q\); Riemann–Roch gives \(\ell(D-E)=\deg(D-E)+1-g\) whenever \(\deg(D-E)>2g-2\).