The sheaf $\OO_X(D)$ and effective divisors as sections

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 $\chi(\OO_X(D))=\deg D+\chi(\OO_X)$.

What is $\OO_X(D)$?

How does an effective divisor give a section of a line bundle?

What is the ideal sheaf of an effective Cartier divisor?

What is the divisor associated to a section of a line bundle?

What is the Riemann--Roch space $L(D)$?

What is the complete linear system associated to a section?

Dependencies

D-5PQ5WWeil and Cartier divisors, and when they agreeuses

Backlinks

PR-DIVLBCartier divisors, invertible sheaves, and the Picard groupuses
D-DIVLINSYSLinear systems, base loci, and base-point freenessuses

What links to this

Line bundles and linear systems