A nonzero section vanishing at a point forces positive degree

Given: A field $k$, an integral proper $k$-scheme $C$ of dimension one, an invertible sheaf $\mathcal{L}$ on $C$ with a nonzero global section $s$ that vanishes at some closed point of $C$.

def-axiom-of-choice. The Axiom of Choice (AC) is the following statement. > Every family of nonempty sets has a choice function > (def-choice-function). Written out: for every set $\mathcal{F}$ all of whose members are nonempty, there exists a function $g$ with domain $\mathcal{F}$ satisfying $g(S)\in S$ for all $S\in \mathcal{F}$. (The Axiom of Choice)

def-degree-invertible-sheaf-proper-dimension-one. Assume the Axiom of Choice, inherited from the Euler-characteristic supplier below (The Axiom of Choice). Let $k$ be a field (def-field) and let $C$ be a proper $k$-scheme (def-proper-morphism) whose underlying topological space is Noetherian of dimension at most one (def-dimension-noetherian-topological-space, def-locally-noetherian-and-noetherian-scheme). (Degree of an invertible sheaf on a proper one-dimensional scheme)

def-integral-scheme. An integral scheme is a nonempty scheme that is reduced and whose underlying topological space is irreducible. Equivalently, it is nonempty and every nonempty affine open is the spectrum of a domain. The latter criterion is independent of the chosen affine open cover. (Integral schemes)

def-invertible-sheaf. Let $X$ be a scheme. An ${\mathcal{O}}_X$-module $\mathcal{L}$ is invertible if it is locally free of rank $1$ (def-locally-free-sheaf-finite-rank): every point $x\in X$ has an open neighbourhood $U$ with $\mathcal{L}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U\mathrm{.}$ Equivalently, $X$ is covered by open sets $U$ on which $\mathcal{L}{\mathrm{\mid }}_U$ admits a generator, that is, a section $s\in \mathcal{L}(U)$ suc (Invertible sheaves)

lem-euler-characteristic-additive-short-exact. Assume the Axiom of Choice, inherited from the finiteness and long-exactness suppliers cited below (The Axiom of Choice). (Euler characteristic is additive in short exact sequences)

lem-euler-characteristic-finite-support-twist-invariance. Assume the Axiom of Choice, inherited from the Euler-characteristic supplier (The Axiom of Choice). Let $k$ be a field (def-field), let $X$ be a proper $k$-scheme, let $p\in X$ be a closed point with residue field $\kappa (p)$ (def-residue-field-scheme-point) and let $i:\mathrm{Spec}\kappa (p)\to X$ be the corresponding closed immersion. (Euler characteristic of a closed point, and invariance under an invertible twist)