Lemma 53.13.8. Let $k$ be a field of characteristic $p > 0$. Let $X$ be a smooth proper curve over $k$. Let $(\mathcal{L}, V)$ be a $\mathfrak g^ r_ d$ with $r \geq 1$. Then one of the following two is true
Proof. Pick two $k$-linearly independent elements $s, t \in V$. Then $f = s/t$ is the rational function defining the morphism $X \to \mathbf{P}^1_ k$ corresponding to the linear series $(\mathcal{L}, ks + kt)$. If this morphism is not generically étale, then $f \in k(X^{(p)})$ by Proposition 53.13.7. Now choose a basis $s_0, \ldots , s_ r$ of $V$ and let $\mathcal{L}' \subset \mathcal{L}$ be the invertible sheaf generated by $s_0, \ldots , s_ r$. Set $f_ i = s_ i/s_0$ in $k(X)$. If for each pair $(s_0, s_ i)$ we have $f_ i \in k(X^{(p)})$, then the morphism
\[ \varphi = \varphi _{(\mathcal{L}', (s_0, \ldots , s_ r)} : X \longrightarrow \mathbf{P}^ r_ k = \text{Proj}(k[T_0, \ldots , T_ r]) \]
factors through $X^{(p)}$ as this is true over the affine open $D_+(T_0)$ and we can extend the morphism over the affine part to the whole of the smooth curve $X^{(p)}$ by Lemma 53.2.2. Introducing notation, say we have the factorization
\[ X \xrightarrow {F_{X/k}} X^{(p)} \xrightarrow {\psi } \mathbf{P}^ r_ k \]
of $\varphi $. Then $\mathcal{N} = \psi ^*\mathcal{O}_{\mathbf{P}^1_ k}(1)$ is an invertible $\mathcal{O}_{X^{(p)}}$-module with $\mathcal{L}' = F_{X/k}^*\mathcal{N}$ and with $\psi ^*T_0, \ldots , \psi ^*T_ r$ $k$-linearly independent (as they pullback to $s_0, \ldots , s_ r$ on $X$). Finally, we have
\[ d = \deg (\mathcal{L}) \geq \deg (\mathcal{L}') = \deg (F_{X/k}) \deg (\mathcal{N}) = p \deg (\mathcal{N}) \]
as desired. Here we used Varieties, Lemmas 33.44.12, 33.44.11, and 33.36.10. $\square$
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