Line bundles of degree at least 2g are base-point-free

Given: A field $k$; a smooth proper geometrically integral curve $C$ over $k$ of genus $g$; an invertible ${\mathcal{O}}_C$-module $L$ with $\mathrm{deg}(L)\ge 2g$; an algebraic closure $\bar{k}$ and the base change $C_{\bar{k}}$.

For an invertible sheaf $\mathcal{E}$ with $\mathrm{deg}(\mathcal{E})>2g-2$ one has $H^1(C,\mathcal{E})=0$ and $h^0(C,\mathcal{E})=\mathrm{deg}(\mathcal{E})+1-g$. (H^1 of a line bundle vanishes above degree 2g - 2, Riemann-Roch in exact form for divisors of degree above 2g - 2)

On the smooth curve $C$ every invertible sheaf is ${\mathcal{O}}_C(D)$ for a divisor $D$ well defined modulo linear equivalence, with $\mathrm{deg}(L)={\mathrm{deg}}_k(D)$; degrees of divisors are additive, ${\mathrm{deg}}_k(D-p)={\mathrm{deg}}_k(D)-[\kappa (p):k]$ for a closed point $p$, and ${\mathcal{O}}_C(D-p)\cong {\mathcal{O}}_C(D)\otimes {\mathcal{O}}_C(-p)$ is the sheaf of $L$ twisted by the point. (Cartier and Weil divisors agree on a smooth curve, Degree divisor proper curve, Invertible sheaf of cartier divisor, Rational sections of line bundles are Cartier divisors)

For a divisor $D$ and a closed point $p$ with residue degree $d=[\kappa (p):k]$ the inclusion ${\mathcal{O}}_C(D)\to {\mathcal{O}}_C(D+p)$ sits in a short exact sequence $0\to {\mathcal{O}}_C(D)\to {\mathcal{O}}_C(D+p)\to i_{\ast }({\mathcal{O}}_C(D+p){\mathrm{\mid }}_p)\to 0$ whose cokernel is the skyscraper at $p$ with $H^0\cong \kappa (p)$ of dimension $d$ and $H^q=0$ for $q\ge 1$; equivalently $0\to L(-p)\to L\to L{\mathrm{\mid }}_p\to 0$ with $H^0(L{\mathrm{\mid }}_p)=\kappa (p)$. (The exact sequence for adding one point to a divisor)

A linear system $\mathrm{\mid }D\mathrm{\mid }$ is base-point-free when no closed point lies in every member, equivalently when its evaluation map is surjective, that is when ${\mathcal{O}}_C(D)$ is globally generated. A base-point-free subspace of $L(D)$ of dimension $r+1\ge 1$ defines a morphism to ${\mathbf{P}}_k^r$ whose pullback of $\mathcal{O}(1)$ is ${\mathcal{O}}_C(D)$; projective space has its standard charts. (Base points and base-point-free linear systems, Complete linear system, Global generation by the evaluation map, A base-point-free linear system defines a morphism to projective space, Relative projective space from standard charts)

For any field extension $K\mathrm{/}k$ and coherent sheaf $\mathcal{F}$ on $C$, $H^q(C_K,{\mathcal{F}}_K)\cong H^q(C,\mathcal{F}){\otimes }_{k}K$ for all $q\ge 0$. Thus the genus and the dimensions of global sections are preserved. The evaluation map on $C_K$ is the base change of the evaluation map on $C$, because the displayed cohomology isomorphism identifies its source. If its cokernel becomes zero over $K$, then on each affine open its module $M$ has $M{\otimes }_{k}K=0$. For a nonzero $M$, the inclusion of a one-dimensional $k$-subspace remains injective after tensoring with the flat $k$-module $K$, and that subspace tensors to $K\mathrm{\ne }0$; hence $M=0$. This proves descent of global generation. (Flat field extension commutes with coherent cohomology, Flat and faithfully flat modules and ring homomorphisms, Modules over a field are projective, flat, and injective, Tensoring is right exact)

Degree after arbitrary field extension. Let $K\mathrm{/}k$ be any field extension and let $x$ be a closed point of $C$, with finite residue field $E=\kappa (x)$. The pullback point scheme is $x_K=\mathrm{Spec}(E{\otimes }_{k}K)$. A finite $k$-basis of $E$ tensors to a $K$-basis, so this finite-dimensional $K$-algebra has dimension $[E:k]$ and is Artinian: a descending chain of ideals is a descending chain of finite-dimensional $K$-subspaces and therefore stabilizes. By the structure theorem for Artinian rings, it is the finite product of its localizations at its maximal ideals. Write $E{\otimes }_{k}K={\prod }_y{A}_y$ over those factors. Each $A_y$ is an Artinian local ring, so its regular module has finite composition length. Every simple factor is its residue field $\kappa (y)$, and additivity of $K$-dimension along that composition series gives ${\dim }_K{A}_y={\mathrm{length}}_{A_y}(A_y)[\kappa (y):K]\mathrm{.}$ The dimension of a finite product is the sum of the dimensions of its factors, so $[E:k]={\sum }_y{\mathrm{length}}_{A_y}(A_y)[\kappa (y):K]\mathrm{.}$ The projection $C_K\to C$ is flat: $K$ is flat over $k$ by Flat and faithfully flat modules and ring homomorphisms, and flatness of morphisms is preserved by base change. The closed point $x$ is an effective Cartier divisor on the smooth curve; its pullback is $x_K$. At each $y$, the local ring of $C_K$ is a DVR, and if a local equation for $x_K$ has order $e$, its quotient has length $e$. Thus the coefficient of $y$ in the pulled-back divisor is ${\mathrm{length}}_{A_y}(A_y)$, and ${\mathrm{deg}}_K(x_K)={\sum }_y{\mathrm{length}}_{A_y}(A_y)[\kappa (y):K]=[E:k]={\mathrm{deg}}_k(x)\mathrm{.}$ By additivity, ${\mathrm{deg}}_K(D_K)={\mathrm{deg}}_k(D)$ for every divisor $D={\sum }_x{n}_x[x]$, with no separability hypothesis and including negative coefficients. Every invertible sheaf is ${\mathcal{O}}_C(D)$ for a divisor $D$ by taking a nonzero rational section. Flat pullback gives ${\mathcal{O}}_{C_K}(D_K)\cong {\mathcal{O}}_C(D{)}_K$; the Cartier/Weil identification and the degree homomorphism on the Picard group therefore give $\mathrm{deg}({\mathcal{F}}_K)=\mathrm{deg}(\mathcal{F})$ for every invertible sheaf $\mathcal{F}$ on $C$. In particular this holds for $K=\bar{k}$. (Degree divisor proper curve, Divisors on a smooth proper curve, Flat morphism of schemes, Flat and faithfully flat modules and ring homomorphisms, Flatness is stable under arbitrary base change, Modules over a field are projective, flat, and injective, Local rings at closed points of smooth curves are discrete valuation rings, An Artinian ring is canonically the finite product of its localizations at its maximal ideals, A commutative ring is Artinian exactly when it has finite length as a module over itself, Length and valuation in a DVR, Rational sections of line bundles are Cartier divisors, Pullback of a Cartier divisor, Pullback of a Cartier divisor computes the pullback of its line bundle, Cartier and Weil divisors agree on a smooth curve, The degree of a divisor descends to the Picard group of a normal proper curve)

The Axiom of Choice: every family of nonempty sets has a choice function. (The Axiom of Choice)