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Tag 0CE9

Chapter 51: Semistable Reduction > Section 51.11: A formula for the genus

Lemma 51.11.8. In Situation 51.9.3 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 20.21.7) 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 51.9.9. 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 51.9.9 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 51.11.5). Thus $\dim_k H^1(C, \mathcal{L}) > 0$ which finishes the proof. $\square$

    The code snippet corresponding to this tag is a part of the file models.tex and is located in lines 5157–5166 (see updates for more information).

    \begin{lemma}
    \label{lemma-genus-reduction-smaller}
    In Situation \ref{situation-regular-model} 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 $(X_k)_{red} \not = X_k$.
    \end{lemma}
    
    \begin{proof}
    By vanishing of cohomology in degrees $\geq 2$ over $X_k$
    (Cohomology, Proposition \ref{cohomology-proposition-vanishing-Noetherian})
    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 \ref{lemma-regular-model-field}. 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 \ref{lemma-regular-model-field}
    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 \ref{lemma-numerical-type-minimal-model}).
    Thus $\dim_k H^1(C, \mathcal{L}) > 0$ which finishes the proof.
    \end{proof}

    Comments (1)

    Comment #2949 by Maciek Zdanowicz on October 10, 2017 a 9:01 pm UTC

    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.

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