Complete linear system

Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field), and let $D$ be a divisor on $C$ (Divisors on a smooth proper curve). The complete linear system of $D$ is the set $\mathrm{\mid }D\mathrm{\mid }=\{D^{\mathrm{\prime }}\text{ effective divisor on }C:D^{\mathrm{\prime }}\text{ is linearly equivalent to }D\},$ the set of effective divisors on $C$ linearly equivalent to $D$; linear equivalence on the closed-point divisor group means that $D^{\mathrm{\prime }}-D$ is the principal divisor of a function in $k(C{)}^{\times }$, and effectivity means nonnegativity of all coefficients (Divisors on a smooth proper curve).

The set in this definition is the set of effective divisors in the linear equivalence class of $D$. This set definition does not assert a projective space structure. For the following correspondence with $P(L(D))$ and the finite-dimensional projective-space structure, assume the Axiom of Choice (The Axiom of Choice): this is the current supplier route for the Cartier/Weil identification, the section dictionary, and proper coherent cohomology finiteness. AC supplies the Dependent Choice premise of Cartier and Weil divisors agree on a smooth curve through AC implies DC implies countable choice.

Under this identification, the relation above is the Cartier relation of Linear equivalence cartier divisors, and coefficientwise effectivity agrees with Cartier effectivity of Effective cartier divisor.

Let $L(D)$ be the Riemann-Roch space of $D$ (The space L(D)), a $k$-subspace of the function field $k(C)$ (Vector space over a field). By Effective divisors linearly equivalent to D are sections modulo scalars the assignment $f\mapsto \mathrm{div}(f)+D$ induces a bijection $(L(D)\setminus \{0\})\mathrm{/}{k}^{\times }⟶\mathrm{\mid }D\mathrm{\mid },[f]⟼\mathrm{div}(f)+D,$ whose inverse sends an effective divisor $D^{\mathrm{\prime }}\in \mathrm{\mid }D\mathrm{\mid }$ to the $k^{\times }$-orbit of a function $f$ with $D^{\mathrm{\prime }}=\mathrm{div}(f)+D$; thus $\mathrm{\mid }D\mathrm{\mid }$ is in bijection with the set of $k$-lines in $L(D)$. One writes $\mathrm{\mid }D\mathrm{\mid }=P(L(D))$ for this set of lines, and calls $\mathrm{\mid }D\mathrm{\mid }$ the complete linear system attached to $D$. It is empty exactly when $L(D)=0$, that is, when no effective divisor is linearly equivalent to $D$ (Effective divisors linearly equivalent to D are sections modulo scalars). For this curve and divisor, $L(D)$ is finite-dimensional by the local coherence route. The curve is finite type over the field $k$, and a field is Noetherian, so every finite-type affine chart of $C$ is Noetherian and $C$ is locally Noetherian. The Cartier construction makes ${\mathcal{O}}_C(D)$ an invertible sheaf, hence locally free of rank one; it is therefore quasi-coherent and of finite type. On a locally Noetherian scheme this makes it coherent. Since $C$ is proper over $k$, the published Finite-dimensional coherent cohomology over a field applies and makes $H^0(C,{\mathcal{O}}_C(D))$ finite-dimensional. The current The space L(D) and rational-section dictionary identify this space with $L(D)$. Thus the set of $k$-lines $P(L(D))$ is the projective space of lines in a finite-dimensional vector space, so the complete linear system carries the structure of a projective linear system.

The construction depends only on the linear equivalence class of $D$. If $D^{\mathrm{\prime }}=D+\mathrm{div}(h)$ for $h\in k(C{)}^{\times }$, then multiplication by $h$ maps $L(D^{\mathrm{\prime }})$ to $L(D)$: for $f\in L(D^{\mathrm{\prime }})$, $\mathrm{div}(hf)+D=\mathrm{div}(f)+\mathrm{div}(h)+D=\mathrm{div}(f)+D^{\mathrm{\prime }}\ge 0.$ Conversely, for $g\in L(D)$, the function $g\mathrm{/}h$ lies in $L(D^{\mathrm{\prime }})$, so this is an isomorphism. Under the two section-to-divisor bijections, the line $[f]$ maps to $[hf]$, and $\mathrm{div}(hf)+D=\mathrm{div}(f)+D^{\mathrm{\prime }}\mathrm{.}$ Thus the associated effective divisor is the same on both sides, and $\mathrm{\mid }D\mathrm{\mid }=\mathrm{\mid }{D}^{\mathrm{\prime }}\mathrm{\mid }$ as sets of effective divisors.

Current supplier interfaces. The effective-Cartier and Cartier-linear- equivalence conventions are given by Effective cartier divisor and Linear equivalence cartier divisors, and the curve-level Cartier and Weil divisors agree on a smooth curve transports them to closed-point divisors while preserving principal divisors. The current The space L(D) body identifies $L(D)$ with $H^0(C,{\mathcal{O}}_C(D))$, and the current Effective divisors linearly equivalent to D are sections modulo scalars body gives the orbit correspondence used above. Finite-dimensionality follows from the local Noetherian/coherence route above and the published Finite-dimensional coherent cohomology over a field. These structural claims use the AC premise stated above; AC supplies the DC premise of the curve Cartier-to-Weil result by AC implies DC implies countable choice. Under the Cartier-to-Weil identification, the sheaf ${\mathcal{O}}_C(D)$ is the invertible sheaf defined by Invertible sheaf of cartier divisor. The set definition remains separate from the projective-space structure.