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

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

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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)