Nontrivial degree-zero line bundles have no sections

The general statement. Let $\mathcal{L}$ be an invertible sheaf on $C$ with ${\mathrm{deg}}_k\mathcal{L}=0$ and $\mathcal{L}\cong ̸{\mathcal{O}}_C$. If $H^0(C,\mathcal{L})\mathrm{\ne }0$, then [F1] gives $\mathcal{L}\cong {\mathcal{O}}_C$, contrary to the hypothesis. Hence $H^0(C,\mathcal{L})=0$.

Base change and factorization. Write $F$ as a homogeneous cubic and pass to $\bar{k}$. By [F7], $E_{\bar{k}}=V_{+}(F)\subseteq {\mathbb{P}}_{\bar{k}}^2$ is smooth. We show that $F$ is irreducible and squarefree over $\bar{k}$. If an irreducible factor $G$ occurs with multiplicity at least two, then $2\mathrm{deg}G\le 3$, so $G$ is linear. At least one coordinate linear form $L$ is not proportional to $G$; thus $G$ and $L$ are coprime. By weighted Bezout [F7], they meet at a closed point $z\in V_{+}(G,L)$. Choose a standard affine chart containing $z$. The dehomogenized equation $f$ of $E_{\bar{k}}$ is divisible by the square of the dehomogenized $G$, so $f\in {\mathfrak{m}}_z^2$ and every first partial derivative of $f$ vanishes at $z$. By [F7] the one-equation chart is not smooth at $z$, contradicting the smoothness of $E_{\bar{k}}$. If $F$ is squarefree but reducible, factor it as $F=GH$ with $G,H$ coprime homogeneous forms of positive degree. Weighted Bezout [F7] gives a closed point $z\in V_{+}(G,H)$. In a standard affine chart containing $z$, the dehomogenized equation is $f=gh\in {\mathfrak{m}}_z^2$, so all its first partial derivatives vanish at $z$. The same Jacobian criterion contradicts smoothness. Thus $F$ is irreducible and squarefree over $\bar{k}$. Since the polynomial ring is a UFD, $(F)$ is prime; consequently $E_{\bar{k}}$ is integral and $E$ is geometrically integral. The given pure dimension one then makes $E$ an integral plane curve.

The degree of ${\mathcal{O}}_E(P-Q)$. Since $P$ and $Q$ are $k$-rational, $[\kappa (P):k]=[\kappa (Q):k]=1$ by [F2]. Therefore ${\mathrm{deg}}_k(P-Q)=1-1=0$. The closed plane subscheme $E$ is proper by [F8], and its smoothness and geometric integrality from step 1.2 make it normal by [F9]. Hence [F2] applies and gives ${\mathrm{deg}}_k{\mathcal{O}}_E(P-Q)=0$.

Properness and genus. By [F8], $E$ is proper as a closed subscheme of projective space. It is smooth by hypothesis and geometrically integral by step 1.2, hence a smooth proper geometrically integral curve. Apply [F5] to the integral plane cubic $E$ to get arithmetic genus one; smoothness and properness identify this with its genus, so $g(E)=1$.

Nontriviality of ${\mathcal{O}}_E(P-Q)$. Suppose ${\mathcal{O}}_E(P-Q)\cong {\mathcal{O}}_E$. By [F3], $P-Q$ is principal, so some $f\in k(E{)}^{\times }$ satisfies $\mathrm{div}(f)=P-Q$. Since $P\mathrm{\ne }Q$, $f$ is nonconstant, and its pole divisor is $(f{)}_{\mathrm{\infty }}=[Q]$, of degree one. By [F4], $f$ defines a finite morphism ${\phi }_f:E\to {\mathbb{P}}_k^1$ whose degree is $[k(E):k(f)]={\mathrm{deg}}_k(f{)}_{\mathrm{\infty }}=1$. Thus the morphism is birational, and [F4] makes it an isomorphism. This contradicts $g(E)=1$ from step 2.2 and $g({\mathbb{P}}_k^1)=0$ from [F6]. Hence ${\mathcal{O}}_E(P-Q)\cong ̸{\mathcal{O}}_E$.

The vanishing for the cubic. By step 2.1 the invertible sheaf ${\mathcal{O}}_E(P-Q)$ has degree zero, and by step 3.1 it is not trivial. Step 1.1 applied to $C=E$ and $\mathcal{L}={\mathcal{O}}_E(P-Q)$ gives $H^0(E,{\mathcal{O}}_E(P-Q))=0$.

Conclusion and Choice accounting. Step 1.1 proves the general statement: a nontrivial degree-zero invertible sheaf on a smooth proper geometrically integral curve has no nonzero global section, by the contrapositive of the section-triviality corollary and without Serre duality. Steps 1.2, 2.1, 2.2, and 3.1 show that every smooth pure-dimension-one plane cubic in the stated scope is a smooth proper geometrically integral genus-one curve and that ${\mathcal{O}}_E(P-Q)$ is degree zero and nontrivial; step 4.1 gives the promised vanishing. AC supplies DC through [F10], and the other inherited Choice uses are exactly those named there. No additional selection is made beyond the given data and the finite nonempty intersections in the Bezout argument.