Lemma 44.6.5. Let $k$ be a separably closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$. Let $K/k$ be a field extension and let $\mathcal{L}$ be an invertible sheaf on $X_ K$. Then there exists an invertible sheaf $\mathcal{L}_0$ on $X$ such that $\dim _ K H^0(X_ K, \mathcal{L} \otimes _{\mathcal{O}_{X_ K}} \mathcal{L}_0|_{X_ K}) = 1$ and $\dim _ K H^1(X_ K, \mathcal{L} \otimes _{\mathcal{O}_{X_ K}} \mathcal{L}_0|_{X_ K}) = 0$.

Proof. This proof is a variant of the proof of Varieties, Lemma 33.43.16. We encourage the reader to read that proof first.

First we pick an ample invertible sheaf $\mathcal{L}_0$ and we replace $\mathcal{L}$ by $\mathcal{L} \otimes _{\mathcal{O}_{X_ K}} \mathcal{L}_0^{\otimes n}|_{X_ K}$ for some $n \gg 0$. The result will be that we may assume that $H^0(X_ K, \mathcal{L}) \not= 0$ and $H^1(X_ K, \mathcal{L}) = 0$. Namely, we will get the vanishing by Cohomology of Schemes, Lemma 30.17.1 and the nonvanishing because the degree of the tensor product is $\gg 0$. We will finish the proof by descending induction on $t = \dim _ K H^0(X_ K, \mathcal{L})$. The base case $t = 1$ is trivial. Assume $t > 1$.

Observe that for a $k$-rational point $x$ of $X$, the inverse image $x_ K$ is a $K$-rational point of $X_ K$. Moreover, there are infinitely many $k$-rational points by Varieties, Lemma 33.25.6. Therefore the points $x_ K$ form a Zariski dense collection of points of $X_ K$.

Let $s \in H^0(X_ K, \mathcal{L})$ be nonzero. From the previous paragraph we deduce there exists a $k$-rational point $x$ such that $s$ does not vanish in $x_ K$. Let $\mathcal{I}$ be the ideal sheaf of $i : x_ K \to X_ K$ as in Varieties, Lemma 33.42.8. Look at the short exact sequence

$0 \to \mathcal{I} \otimes _{\mathcal{O}_{X_ K}} \mathcal{L} \to \mathcal{L} \to i_*i^*\mathcal{L} \to 0$

Observe that $H^0(X_ K, i_*i^*\mathcal{L}) = H^0(x_ K, i^*\mathcal{L})$ has dimension $1$ over $K$. Since $s$ does not vanish at $x$ we conclude that

$H^0(X_ K, \mathcal{L}) \longrightarrow H^0(X, i_*i^*\mathcal{L})$

is surjective. Hence $\dim _ K H^0(X_ K, \mathcal{I} \otimes _{\mathcal{O}_{X_ K}} \mathcal{L}) = t - 1$. Finally, the long exact sequence of cohomology also shows that $H^1(X_ K, \mathcal{I} \otimes _{\mathcal{O}_{X_ K}} \mathcal{L}) = 0$ thereby finishing the proof of the induction step. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).