Lemma 55.11.8. In Situation 55.9.3 assume $X$ is a minimal model, $\gcd (m_1, \ldots , m_ n) = 1$, and $H^0((X_ k)_{red}, \mathcal{O}) = k$. Then the map

$H^1(X_ k, \mathcal{O}_{X_ k}) \to H^1((X_ k)_{red}, \mathcal{O}_{(X_ k)_{red}})$

is surjective and has a nontrivial kernel as soon as $(X_ k)_{red} \not= X_ k$.

Proof. By vanishing of cohomology in degrees $\geq 2$ over $X_ k$ (Cohomology, Proposition 20.20.7) any surjection of abelian sheaves on $X_ k$ induces a surjection on $H^1$. Consider the sequence

$(X_ k)_{red} = Z_0 \subset Z_1 \subset \ldots \subset Z_ m = X_ k$

of Lemma 55.9.9. Since the field maps $H^0(Z_ j, \mathcal{O}_{Z_ j}) \to H^0((X_ k)_{red}, \mathcal{O}_{(X_ k)_{red}}) = k$ are injective we conclude that $H^0(Z_ j, \mathcal{O}_{Z_ j}) = k$ for $j = 0, \ldots , m$. It follows that $H^0(X_ k, \mathcal{O}_{X_ k}) \to H^0(Z_{m - 1}, \mathcal{O}_{Z_{m - 1}})$ is surjective. Let $C = C_{i_ m}$. Then $X_ k = Z_{m - 1} + C$. Let $\mathcal{L} = \mathcal{O}_ X(-Z_{m - 1})|_ C$. Then $\mathcal{L}$ is an invertible $\mathcal{O}_ C$-module. As in the proof of Lemma 55.9.9 there is an exact sequence

$0 \to \mathcal{L} \to \mathcal{O}_{X_ k} \to \mathcal{O}_{Z_{m - 1}} \to 0$

of coherent sheaves on $X_ k$. We conclude that we get a short exact sequence

$0 \to H^1(C, \mathcal{L}) \to H^1(X_ k, \mathcal{O}_{X_ k}) \to H^1(Z_{m - 1}, \mathcal{O}_{Z_{m - 1}}) \to 0$

The degree of $\mathcal{L}$ on $C$ over $k$ is

$(C \cdot -Z_{m - 1}) = (C \cdot C - X_ k) = (C \cdot C)$

Set $\kappa = H^0(C, \mathcal{O}_ C)$ and $w = [\kappa : k]$. By definition of the degree of an invertible sheaf we see that

$\chi (C, \mathcal{L}) = \chi (C, \mathcal{O}_ C) + (C \cdot C) = w(1 - g_ C) + (C \cdot C)$

where $g_ C$ is the genus of $C$. This expression is $< 0$ as $X$ is minimal and hence $C$ is not an exceptional curve of the first kind (see proof of Lemma 55.11.5). Thus $\dim _ k H^1(C, \mathcal{L}) > 0$ which finishes the proof. $\square$

Comment #2949 by Maciek Zdanowicz on

This is just a little typo, but there's $K$ instead of $k$ in the formula $H^1(X_k, \mathcal{O}_{X_k}) \to H^1((X_k)_{red}, \mathcal{O}_{(X_K)_{red}})$ in the statement of the lemma.

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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CE9. Beware of the difference between the letter 'O' and the digit '0'.