The Stacks project

Lemma 53.18.2. Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Then

  1. We have $g_{geom}(X/k) = g_{geom}(X_{red}/k)$.

  2. If $X' \to X$ is a birational proper morphism, then $g_{geom}(X'/k) = g_{geom}(X/k)$.

  3. 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). $\square$

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