Lemma 53.18.1. Let k be a field. Let X be a proper scheme of dimension \leq 1 over k. Then
g_{geom}(X/k) = \sum \nolimits _{C \subset X} g_{geom}(C/k)
where the sum is over irreducible components C \subset X of dimension 1.
53.18 Genus versus geometric genus
Let k be a field with algebraic closure \overline{k}. Let X be a proper scheme of dimension \leq 1 over k. We define g_{geom}(X/k) to be the sum of the geometric genera of the irreducible components of X_{\overline{k}} which have dimension 1.
Proof. This is immediate from the definition and the fact that an irreducible component \overline{Z} of X_{\overline{k}} maps onto an irreducible component Z of X (Varieties, Lemma 33.8.10) of the same dimension (Morphisms, Lemma 29.28.3 applied to the generic point of \overline{Z}). \square
Lemma 53.18.2. Let k be a field. Let X be a proper scheme of dimension \leq 1 over k. Then
We have g_{geom}(X/k) = g_{geom}(X_{red}/k).
If X' \to X is a birational proper morphism, then g_{geom}(X'/k) = g_{geom}(X/k).
If X^\nu \to X is the normalization morphism, then g_{geom}(X^\nu /k) = g_{geom}(X/k).
Proof. Part (1) is immediate from Lemma 53.18.1. If X' \to X is proper birational, then it is finite and an isomorphism over a dense open (see Varieties, Lemmas 33.17.2 and 33.17.3). Hence X'_{\overline{k}} \to X_{\overline{k}} is an isomorphism over a dense open. Thus the irreducible components of X'_{\overline{k}} and X_{\overline{k}} are in bijective correspondence and the corresponding components have isomorphic function fields. In particular these components have isomorphic nonsingular projective models and hence have the same geometric genera. This proves (2). Part (3) follows from (1) and (2) and the fact that X^\nu \to X_{red} is birational (Morphisms, Lemma 29.54.7). \square
Lemma 53.18.3. Let k be a field. Let X be a proper scheme of dimension \leq 1 over k. Let f : Y \to X be a finite morphism such that there exists a dense open U \subset X over which f is a closed immersion. Then
\dim _ k H^1(X, \mathcal{O}_ X) \geq \dim _ k H^1(Y, \mathcal{O}_ Y)
Proof. Consider the exact sequence
0 \to \mathcal{G} \to \mathcal{O}_ X \to f_*\mathcal{O}_ Y \to \mathcal{F} \to 0
of coherent sheaves on X. By assumption \mathcal{F} is supported in finitely many closed points and hence has vanishing higher cohomology (Varieties, Lemma 33.33.3). On the other hand, we have H^2(X, \mathcal{G}) = 0 by Cohomology, Proposition 20.20.7. It follows formally that the induced map H^1(X, \mathcal{O}_ X) \to H^1(X, f_*\mathcal{O}_ Y) is surjective. Since H^1(X, f_*\mathcal{O}_ Y) = H^1(Y, \mathcal{O}_ Y) (Cohomology of Schemes, Lemma 30.2.4) we conclude the lemma holds. \square
Lemma 53.18.4. Let k be a field. Let X be a proper scheme of dimension \leq 1 over k. If X' \to X is a birational proper morphism, then
\dim _ k H^1(X, \mathcal{O}_ X) \geq \dim _ k H^1(X', \mathcal{O}_{X'})
If X is reduced, H^0(X, \mathcal{O}_ X) \to H^0(X', \mathcal{O}_{X'}) is surjective, and equality holds, then X' = X.
Proof. If f : X' \to X is proper birational, then it is finite and an isomorphism over a dense open (see Varieties, Lemmas 33.17.2 and 33.17.3). Thus the inequality by Lemma 53.18.3. Assume X is reduced. Then \mathcal{O}_ X \to f_*\mathcal{O}_{X'} is injective and we obtain a short exact sequence
0 \to \mathcal{O}_ X \to f_*\mathcal{O}_{X'} \to \mathcal{F} \to 0
Under the assumptions given in the second statement, we conclude from the long exact cohomology sequence that H^0(X, \mathcal{F}) = 0. Then \mathcal{F} = 0 because \mathcal{F} is generated by global sections (Varieties, Lemma 33.33.3). and \mathcal{O}_ X = f_*\mathcal{O}_{X'}. Since f is affine this implies X = X'. \square
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
Lemma 53.18.6. Let k be a field. Let X be a proper scheme of dimension \leq 1 over k. Let \ell be a prime number invertible in k. Then
\dim _{\mathbf{F}_\ell } \mathop{\mathrm{Pic}}\nolimits (X)[\ell ] \leq \dim _ k H^1(X, \mathcal{O}_ X) + g_{geom}(X/k)
where g_{geom}(X/k) is as defined above.
Proof. The map \mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (X_{\overline{k}}) is injective by Varieties, Lemma 33.30.3. By Cohomology of Schemes, Lemma 30.5.2 \dim _ k H^1(X, \mathcal{O}_ X) equals \dim _{\overline{k}} H^1(X_{\overline{k}}, \mathcal{O}_{X_{\overline{k}}}). Hence we may assume k is algebraically closed.
Let X_{red} be the reduction of X. Then the surjection \mathcal{O}_ X \to \mathcal{O}_{X_{red}} induces a surjection H^1(X, \mathcal{O}_ X) \to H^1(X, \mathcal{O}_{X_{red}}) because cohomology of quasi-coherent sheaves vanishes in degrees \geq 2 by Cohomology, Proposition 20.20.7. Since X_{red} \to X induces an isomorphism on irreducible components over \overline{k} and an isomorphism on \ell -torsion in Picard groups (Picard Schemes of Curves, Lemma 44.7.2) we may replace X by X_{red}. In this way we reduce to Proposition 53.17.3. \square
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