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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