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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