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

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).