The Stacks project

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 $1$.

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 $\overline{Z}$). $\square$

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