A nontrivial degree-zero line bundle has no nonzero section

Given: An algebraically closed field $k$, the smooth plane cubic $E=V_{+}(F)$ with $F=y^{2}z-x^3-ax{z}^2-b{z}^3$, and distinct closed points $P,Q\in E$.

Let $X=V_{+}(G)\subseteq {\mathbb{P}}_k^2$ be cut out by a nonzero homogeneous form of degree $d\ge 1$ and assume $X$ is a curve (integral of dimension one); then $H^0(X,{\mathcal{O}}_X)=k$ and $p_a(X)=1-\chi ({\mathcal{O}}_X)=\frac{(d-1)(d-2)}{2}$. In particular a smooth plane cubic is an integral proper curve with $p_a=1$, and a line has $p_a=0$. (Arithmetic genus of a plane curve)

A curve over $k$ is geometrically integral, separated, finite type of chain dimension one; a smooth proper curve is in particular integral and reduced, and the arithmetic genus $p_a(X)=1-\chi ({\mathcal{O}}_X)$ of an integral proper curve is an invariant of its isomorphism class, because the cohomology of the structure sheaf depends only on the scheme up to isomorphism. (Curves over a field, Genus and arithmetic genus of a curve)

On a smooth curve divisors are finite $\mathbb{Z}$-combinations of closed points, ${\mathrm{deg}}_k({\sum }_x{n}_x[x])={\sum }_x{n}_x[\kappa (x):k]$, and over an algebraically closed field every closed point has residue field $k$ and degree one. (Divisors on a smooth proper curve, Degree divisor proper curve)

For a smooth proper geometrically integral curve $C$: the group $\mathrm{Pic}(C)$ of isomorphism classes of invertible sheaves is identified with the divisor class group by $[D]\mapsto [{\mathcal{O}}_C(D)]$, the degree ${\mathrm{deg}}_k$ descends to a homomorphism $\mathrm{Pic}(C)\to \mathbb{Z}$ taking ${\mathcal{O}}_C(D)$ to ${\mathrm{deg}}_{k}D$, and ${\mathcal{O}}_C(D)\cong {\mathcal{O}}_C(D^{\mathrm{\prime }})$ if and only if $D$ and $D^{\mathrm{\prime }}$ are linearly equivalent; the sheaf ${\mathcal{O}}_C(D)$ is the invertible sheaf attached to the Cartier divisor $D$. (Cartier and Weil divisors agree on a smooth curve, The degree of a divisor descends to the Picard group of a normal proper curve, Invertible sheaf of cartier divisor)

A nonzero rational section $s$ of an invertible sheaf $\mathcal{M}$ on a smooth curve $C$ determines a divisor $\mathrm{div}(s)$ with $\mathcal{M}\cong {\mathcal{O}}_C(\mathrm{div}(s))$ carrying the canonical section $1$ to $s$; consequently nonzero global sections of $\mathcal{M}$ correspond to effective divisors $D$ with ${\mathcal{O}}_C(D)\cong \mathcal{M}$. (Rational sections of line bundles are Cartier divisors, Effective divisors linearly equivalent to D are sections modulo scalars)

If $D$ is an effective divisor on a proper geometrically integral curve over $k$, then ${\mathrm{deg}}_{k}D\ge 0$, and ${\mathrm{deg}}_{k}D=0$ if and only if $D=0$. (Effective divisors have nonnegative degree)

A nonconstant rational function $f\in k(C{)}^{\times }$ on a smooth proper geometrically integral curve defines a finite morphism ${\phi }_f:C\to {\mathbb{P}}_k^1$ whose degree is $[k(C):k(f)]$ and whose fibre over infinity is the pole divisor $(f{)}_{\mathrm{\infty }}$, of the same degree. (A nonconstant rational function defines a finite map to the projective line, Degree of a nonconstant morphism of curves)

Under Choice the assignment $f\mapsto f^{\ast }$ is a bijection from dominant morphisms between smooth proper geometrically integral curves over $k$ onto injective $k$-algebra homomorphisms of function fields, and every birational rational map between smooth proper geometrically integral curves is represented by a $k$-isomorphism. (Smooth proper curves, dominant morphisms and function fields, Birational smooth proper curves are isomorphic)

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

Polynomial rings in finitely many variables over a field are UFDs; irreducibles are prime. A nonzero positive-degree plane hypersurface has pure dimension one. Closed immersions and projective-space structure maps are proper, and proper morphisms compose. (Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes, Nontrivial projective hypersurface sections, Finite-dimensional projective space is proper over every base, Closed immersions are proper, Properness survives composition)