Lemma 53.13.9. Let $k$ be a field. Let $X$ be a smooth proper curve over $k$ with $H^0(X, \mathcal{O}_ X) = k$ and genus $g \geq 2$. Then there exists a closed point $x \in X$ with $\kappa (x)/k$ separable of degree $\leq 2g - 2$.

Proof. Set $\omega = \Omega _{X/k}$. By Lemma 53.8.4 this has degree $2g - 2$ and has $g$ global sections. Thus we have a $\mathfrak g^{g - 1}_{2g - 2}$. By the trivial Lemma 53.3.3 there exists a $\mathfrak g^1_{2g - 2}$ and by Lemma 53.3.4 we obtain a morphism

$\varphi : X \longrightarrow \mathbf{P}^1_ k$

of some degree $d \leq 2g - 2$. Since $\varphi$ is flat (Lemma 53.2.3) and finite (Lemma 53.2.4) it is finite locally free of degree $d$ (Morphisms, Lemma 29.48.2). Pick any rational point $t \in \mathbf{P}^1_ k$ and any point $x \in X$ with $\varphi (x) = t$. Then

$d \geq [\kappa (x) : \kappa (t)] = [\kappa (x) : k]$

for example by Morphisms, Lemmas 29.56.3 and 29.56.2. Thus if $k$ is perfect (for example has characteristic zero or is finite) then the lemma is proved. Thus we reduce to the case discussed in the next paragraph.

Assume that $k$ is an infinite field of characteristic $p > 0$. As above we will use that $X$ has a $\mathfrak g^{g - 1}_{2g - 2}$. The smooth proper curve $X^{(p)}$ has the same genus as $X$. Hence its genus is $> 0$. We conclude that $X^{(p)}$ does not have a $\mathfrak g^{g - 1}_ d$ for any $d \leq g - 1$ by Lemma 53.3.5. Applying Lemma 53.13.8 to our $\mathfrak g^{g - 1}_{2g - 2}$ (and noting that $2g - 2/p \leq g - 1$) we conclude that possibility (2) does not occur. Hence we obtain a morphism

$\varphi : X \longrightarrow \mathbf{P}^1_ k$

which is generically étale (in the sense of the lemma) and has degree $\leq 2g - 2$. Let $U \subset X$ be the nonempty open subscheme where $\varphi$ is étale. Then $\varphi (U) \subset \mathbf{P}^1_ k$ is a nonempty Zariski open and we can pick a $k$-rational point $t \in \varphi (U)$ as $k$ is infinite. Let $u \in U$ be a point with $\varphi (u) = t$. Then $\kappa (u)/\kappa (t)$ is separable (Morphisms, Lemma 29.36.7), $\kappa (t) = k$, and $[\kappa (u) : k] \leq 2g - 2$ as before. $\square$

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