Lemma 53.18.5. Let $k$ be a field. Let $C$ be a proper curve over $k$. Set $\kappa = H^0(C, \mathcal{O}_ C)$. Then
\[ [\kappa : k]_ s \dim _\kappa H^1(C, \mathcal{O}_ C) \geq g_{geom}(C/k) \]
Proof. Varieties, Lemma 33.26.2 implies $\kappa $ is a field and a finite extension of $k$. By Fields, Lemma 9.14.8 we have $[\kappa : k]_ s = |\mathop{\mathrm{Mor}}\nolimits _ k(\kappa , \overline{k})|$ and hence $\mathop{\mathrm{Spec}}(\kappa \otimes _ k \overline{k})$ has $[\kappa : k]_ s$ points each with residue field $\overline{k}$. Thus
\[ C_{\overline{k}} = \bigcup \nolimits _{t \in \mathop{\mathrm{Spec}}(\kappa \otimes _ k \overline{k})} C_ t \]
(set theoretic union). Here $C_ t = C \times _{\mathop{\mathrm{Spec}}(\kappa ), t} \mathop{\mathrm{Spec}}(\overline{k})$ where we view $t$ as a $k$-algebra map $t : \kappa \to \overline{k}$. The conclusion is that $g_{geom}(C/k) = \sum _ t g_{geom}(C_ t/\overline{k})$ and the sum is over an index set of size $[\kappa : k]_ s$. We have
\[ H^0(C_ t, \mathcal{O}_{C_ t}) = \overline{k} \quad \text{and}\quad \dim _{\overline{k}} H^1(C_ t, \mathcal{O}_{C_ t}) = \dim _\kappa H^1(C, \mathcal{O}_ C) \]
by cohomology and base change (Cohomology of Schemes, Lemma 30.5.2). Observe that the normalization $C_ t^\nu $ is the disjoint union of the nonsingular projective models of the irreducible components of $C_ t$ (Morphisms, Lemma 29.54.6). Hence $\dim _{\overline{k}} H^1(C_ t^\nu , \mathcal{O}_{C_ t^\nu })$ is equal to $g_{geom}(C_ t/\overline{k})$. By Lemma 53.18.3 we have
\[ \dim _{\overline{k}} H^1(C_ t, \mathcal{O}_{C_ t}) \geq \dim _{\overline{k}} H^1(C_ t^\nu , \mathcal{O}_{C_ t^\nu }) \]
and this finishes the proof. $\square$
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