The Stacks project

[Lemma 2.6, Artin-Winters]

Lemma 55.9.10. In Situation 55.9.3 let $d = \gcd (m_1, \ldots , m_ n)$. Let $D = \sum (m_ i/d) C_ i$ as an effective Cartier divisor on $X$. Then

\[ 1 - g_ C = d [\kappa : k] (1 - g_ D) \]

where $g_ C$ is the genus of $C$, $g_ D$ is the genus of $D$, and $\kappa = H^0(D, \mathcal{O}_ D)$.

Proof. By Lemma 55.9.9 we see that $\kappa $ is a field and a finite extension of $k$. Since also $H^0(C, \mathcal{O}_ C) = K$ we see that the genus of $C$ and $D$ are defined (see Algebraic Curves, Definition 53.8.1) and we have $g_ C = \dim _ K H^1(C, \mathcal{O}_ C)$ and $g_ D = \dim _\kappa H^1(D, \mathcal{O}_ D)$. By Derived Categories of Schemes, Lemma 36.32.2 we have

\[ 1 - g_ C = \chi (C, \mathcal{O}_ C) = \chi (X_ k, \mathcal{O}_{X_ k}) = \dim _ k H^0(X_ k, \mathcal{O}_{X_ k}) - \dim _ k H^1(X_ k, \mathcal{O}_{X_ k}) \]

We claim that

\[ \chi (X_ k, \mathcal{O}_{X_ k}) = d \chi (D, \mathcal{O}_ D) \]

This will prove the lemma because

\[ \chi (D, \mathcal{O}_ D) = \dim _ k H^0(D, \mathcal{O}_ D) - \dim _ k H^1(D, \mathcal{O}_ D) = [\kappa : k](1 - g_ D) \]

Observe that $X_ k = dD$ as an effective Cartier divisor. To prove the claim we prove by induction on $1 \leq r \leq d$ that $\chi (rD, \mathcal{O}_{rD}) = r \chi (D, \mathcal{O}_ D)$. The base case $r = 1$ is trivial. If $1 \leq r < d$, then we consider the short exact sequence

\[ 0 \to \mathcal{O}_ X(rD)|_ D \to \mathcal{O}_{(r + 1)D} \to \mathcal{O}_{rD} \to 0 \]

of Divisors, Lemma 31.14.3. By additivity of Euler characteristics (Varieties, Lemma 33.33.2) it suffices to prove that $\chi (D, \mathcal{O}_ X(rD)|_ D) = \chi (D, \mathcal{O}_ D)$. This is true because $\mathcal{O}_ X(rD)|_ D$ is a torsion element of $\mathop{\mathrm{Pic}}\nolimits (D)$ (Lemma 55.9.8) and because the degree of a line bundle is additive (Varieties, Lemma 33.44.7) hence zero for torsion invertible sheaves. $\square$

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