Degree 2g-1 does not force base-point-freeness

Given: the Axiom of Choice and its consequence Dependent Choice; a field $k$, a smooth proper geometrically integral curve $C$ of genus $g\ge 1$ over $k$ with a $k$-rational point $p$, and the invertible sheaf $\mathcal{L}={\mathcal{O}}_C(K_C+p)$ with associated divisor $D=K_C+p$.

${\mathrm{deg}}_k{K}_C=2g-2$, so ${\mathrm{deg}}_{k}D=2g-1$; and the divisor--invertible-sheaf dictionary identifies $\mathcal{L}(-p)={\mathcal{O}}_C(K_C)$ and $H^0(C,\mathcal{L}(-p))\subseteq H^0(C,\mathcal{L})$. (The canonical divisor has degree 2g - 2, Canonical bundle and canonical divisors, Degree divisor proper curve, Invertible sheaf of cartier divisor, Cartier and Weil divisors agree on a smooth curve)

For a divisor of degree $>2g-2$ on a curve of genus $g$ the nonspecial formula gives $\mathrm{\ell }(D)={\mathrm{deg}}_{k}D+1-g$ and $H^1=0$; equivalently $\mathcal{L}$ has $h^0(C,\mathcal{L})=\mathrm{deg}\mathcal{L}+1-g$. (Riemann-Roch in exact form for divisors of degree above 2g - 2, H^1 of a line bundle vanishes above degree 2g - 2, The full Riemann-Roch theorem for divisors on a smooth proper curve, The Riemann-Roch dimension l(D))

$\mathrm{\ell }(K_C)=h^0(C,{\omega }_C)=g$, where ${\omega }_C={\mathcal{O}}_C(K_C)$ is the canonical bundle; the Riemann-Roch space of $K_C$ is the space of holomorphic differentials. (The canonical bundle has exactly g independent sections, The space L(D))

A closed point $q$ is a base point of the complete linear system $\mathrm{\mid }D\mathrm{\mid }$ of a divisor $D$ exactly when every global section of $\mathcal{O}(D)$ vanishes at $q$, equivalently $H^0(C,\mathcal{O}(D-q))=H^0(C,\mathcal{O}(D))$; the sheaf is base-point-free when no closed point is a base point. (Base points and base-point-free linear systems)

A line bundle of degree at least $2g$ on a curve of genus $g$ is base-point-free; this is the statement whose degree bound is tested here. (Line bundles of degree at least 2g are base-point-free)

On a genus-one curve, the canonical bundle is trivial and every canonical divisor is principal; hence $K_C\sim 0$ and ${\mathcal{O}}_C(K_C+p)\cong {\mathcal{O}}_C(p)$. The divisor-line-bundle dictionary identifies this linear equivalence with the corresponding isomorphism of invertible sheaves. (The canonical bundle of a genus-one curve is trivial, Rational sections of line bundles are Cartier divisors, Cartier and Weil divisors agree on a smooth curve)

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

In ZF, the Axiom of Choice implies Dependent Choice; this supplies the Dependent Choice premise of the Cartier-to-Weil divisor dictionary used in [F1] and [F6]. (AC implies DC implies countable choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)