Complete linear system of a divisor ($|D|$)

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.

The Iitaka dimension of a Cartier divisor $D$ is the maximum dimension of the images of the rational maps given by its nonempty multiples $|mD|$. It is $-\infty$ if all positive multiples have no sections. For an integral $n$-dimensional projective variety it lies between zero and $n$ when nonnegative. The Kodaira dimension of the variety is the special case $D=K_X$.

The movable part is $|D-F|$, where $F$ is the fixed part. It has no fixed components. On a smooth algebraic surface its associated divisor is nef: for each curve choose a member avoiding that curve and use nonnegative local intersection multiplicities.

For a nonempty complete linear system of a divisor on a smooth projective variety, the coefficient of a prime divisor in its fixed part is the minimum of its multiplicities in all members. Multiplication by the section of $F$ identifies the sections of $D-F$ with those of $D$.

A fixed component is a prime divisor appearing in every member of a complete linear system of a divisor. This concerns divisorial components; a linear system can have isolated basepoints even when it has no fixed component.

A divisor is basepoint-free when its global sections have no common zero. Its complete linear system therefore defines a Kodaira map everywhere.

For an invariant Cartier divisor $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$ on a complete smooth toric variety, let $m_{\sigma }$ satisfy $⟨m_{\sigma },v_{\rho }⟩=-a_{\rho }$ for rays in a maximal cone $\sigma$. The divisor is basepoint-free exactly when every $m_{\sigma }$ lies in the lattice polytope of a toric divisor $P_D$. Its character section then trivializes the line bundle on the entire chart $U_{\sigma }$.

// Target: geometry-and-topology.bigb

A very ample divisor is a divisor whose associated line bundle is a very ample line bundle. A divisor is very ample when its complete linear system defines a closed embedding into projective space.

On a smooth projective curve of genus $g$, every divisor of degree at least $2g+1$ is very ample. Riemann--Roch shows that its sections separate distinct points and tangent directions by comparing $L(D)$ with $L(D-P-Q)$ and $L(D-2P)$.