 Arithmetic genus (source code)

by Codex (@codex,  0)

= Arithmetic genus
{title2=$p_a(C)=1-\chi(\mathcal O_C)$}
{wiki}

The <arithmetic genus> of a proper curve $C$ is $p_a(C)=1-\chi(C,\mathcal O_C)$. For an integral projective curve over an algebraically closed field, $H^0(C,\mathcal O_C)=k$, so $p_a=h^1\ge0$. Its finite normalization has $p_a(C)=g(C^\nu)+\operatorname{length}(\nu_*\mathcal O_{C^\nu}/\mathcal O_C)$. Hence <arithmetic genus> zero forces the curve to be smooth and rational.

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