Base points and base-point-free linear systems

Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field) with function field $k(C)$, let $D$ be a divisor on $C$ (Divisors on a smooth proper curve), and let $V\subseteq L(D)$ be a $k$-subspace of the Riemann-Roch space (The space L(D)).

Write $D={\sum }_x{n}_x[x]$, so that $n_x={\mathrm{ord}}_x(D)$ is the coefficient of $D$ at the closed point $x$. A nonzero $f\in L(D)$ satisfies $\mathrm{div}(f)+D\ge 0$ by definition of $L(D)$, and the effective divisor $\mathrm{div}(f)+D$ depends only on the $k^{\times }$-orbit of $f$ (The space L(D)).

Assume the Axiom of Choice for the current local-order, Cartier/Weil, and finite-dimensionality supplier routes (The Axiom of Choice). It supplies Dependent Choice through AC implies DC implies countable choice, as used by the current Cartier/Weil route. The pointwise vanishing and base-point conditions themselves are the displayed divisor inequalities.

By Cartier and Weil divisors agree on a smooth curve the divisor $D$ is a Cartier divisor on the curve and carries an associated invertible sheaf ${\mathcal{O}}_C(D)$ (Invertible sheaf of cartier divisor), and by the promised identification of The space L(D) the space $L(D)$ is the space of global sections of that sheaf: a nonzero $f\in L(D)$ corresponds to the global section $s_f$ whose divisor is $\mathrm{div}(s_f)=\mathrm{div}(f)+D\mathrm{.}$ With this dictionary in place, for a closed point $x$ of $C$:

1. a nonzero $f\in V$ vanishes at $x$ when $x$ lies in the divisor $\mathrm{div}(f)+D$, that is, when ${\mathrm{ord}}_x(f)+n_x\ge 1$; equivalently, the section $s_f$ has zero value in the fibre of ${\mathcal{O}}_C(D)$ at $x$. This is a condition on the section of ${\mathcal{O}}_C(D)$, not on the rational function $f$ alone: it differs from the naive condition ${\mathrm{ord}}_x(f)\ge 1$ whenever $n_x\mathrm{\ne }0$, since the section-vanishing threshold is ${\mathrm{ord}}_x(f)\ge 1-n_x$; for $n_x>0$ this can hold even if $f$ does not vanish as a rational function, while for $n_x<0$ it requires a higher-order zero than the naive test;
2. $x$ is a base point of $V$ when every nonzero $f\in V$ vanishes at $x$, i.e. when $x$ belongs to the support of $\mathrm{div}(f)+D$ for every nonzero $f\in V$;
3. the linear system $P(V)$ attached to $V$ is the image of $V\setminus \{0\}$ in $\mathrm{\mid }D\mathrm{\mid }=P(L(D))$ under $f\mapsto \mathrm{div}(f)+D$ (Complete linear system), namely the set of effective divisors $\mathrm{div}(f)+D$ with $f\in V\setminus \{0\}$; by the previous two clauses, $x$ is a base point of $V$ if and only if every divisor of $P(V)$ contains $x$, that is, if and only if the whole subsystem $P(V)$ passes through $x$.

The subspace $V$ is base-point-free when it has no base point. The complete linear system $\mathrm{\mid }D\mathrm{\mid }$ is base-point-free when $L(D)$ is base-point-free as a subspace of itself, and a divisor $D$, or the line bundle ${\mathcal{O}}_C(D)$, is called base-point-free when $\mathrm{\mid }D\mathrm{\mid }$ is base-point-free.

For the finite-basis evaluation formulation, $L(D)$ is finite-dimensional by the following 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 ${\mathcal{O}}_C(D)$ invertible, hence locally free of rank one; it is therefore quasi-coherent and of finite type, and thus coherent on the locally Noetherian scheme $C$. Proper cohomology finiteness Finite-dimensional coherent cohomology over a field makes $H^0(C,{\mathcal{O}}_C(D))$ finite-dimensional. The section dictionary above identifies this space with $L(D)$, so every subspace $V\subseteq L(D)$ is finite-dimensional. Under the Axiom of Choice already assumed, put $m={\dim }_{k}V$ and choose a basis $f_1,\dots ,f_m$ (the empty basis when $m=0$); let $s_i$ be the section corresponding to $f_i$, and define ${\mathrm{ev}}_V:{\mathcal{O}}_C^m⟶{\mathcal{O}}_C(D),(g_1,\dots ,g_m)⟼{\sum }_{i=1}^m{g}_i{s}_i\mathrm{.}$ If $m=0$, this is the zero morphism and is not surjective, since $C$ is nonempty and ${\mathcal{O}}_C(D)$ has nonzero rank-one stalks. For $m>0$, at a closed point $x$ the stalk map is surjective exactly when some $s_i$ has nonzero image in the fibre: in a local frame its image is generated by the coefficients of the $s_i$, and these generate the local ring exactly when one coefficient is a unit, equivalently has nonzero residue. Thus:

• $x$ is a base point of $V$ if and only if ${\mathrm{ev}}_V$ fails to be surjective on stalks at $x$;
• $V$ is base-point-free if and only if ${\mathrm{ev}}_V$ is surjective; equivalently, the subsheaf of ${\mathcal{O}}_C(D)$ generated by the images of $s_1,\dots ,s_m$ is all of ${\mathcal{O}}_C(D)$, i.e. ${\mathcal{O}}_C(D)$ is globally generated by $V$ in the sense of Global generation by the evaluation map (the notion does not depend on the chosen basis). Indeed, by Proper closed subsets of a curve are finite every point of this integral one-dimensional curve is closed or generic. If $m>0$, a basis element is a nonzero rational function, so its corresponding section has nonzero generic value by the section dictionary. Therefore surjectivity at all closed points also gives surjectivity at the generic point; the converse follows by restricting a surjective sheaf map to stalks;
• for the complete system $V=L(D)$, put $m={\dim }_{k}L(D)$. Then $\mathrm{\mid }D\mathrm{\mid }$ is base-point-free exactly when the evaluation map ${\mathcal{O}}_C^m\to {\mathcal{O}}_C(D)$ is surjective, with zero source if $m=0$; equivalently, exactly when ${\mathcal{O}}_C(D)$ is globally generated by these sections. When $m>0$, writing $r+1=m$ recovers the usual indexed basis notation.

The degenerate subspace $V=0$ has no nonzero element, so every closed point $x$ is vacuously a base point of $V$. The curve has a closed point: a nonempty proper irreducible closed subset occurs in the strict chain witnessing its dimension one, and Proper closed subsets of a curve are finite says its points are closed. Thus $V$ is not base-point-free, in agreement with the nonsurjective rank-zero evaluation map. The associated morphism of the next item is therefore only asserted for base-point-free systems of dimension $r+1\ge 1$.

The supplier interfaces used here are present in the current item bodies: Cartier and Weil divisors agree on a smooth curve identifies the curve divisor with a Cartier divisor, Invertible sheaf of cartier divisor defines ${\mathcal{O}}_C(D)$, The space L(D) identifies $L(D)$ with $H^0(C,{\mathcal{O}}_C(D))$ in $k(C)$, and Rational sections of line bundles are Cartier divisors gives the section-divisor dictionary. The finite-basis use follows from the local Noetherian/coherence argument above and the published Finite-dimensional coherent cohomology over a field; AC supplies its choice premise and the DC premise of the Cartier-to-Weil interface. The earlier “not yet authored” supplier notice is stale; the current bodies supply the remaining interfaces used above.