= Genus of a complete-intersection space curve
{title2=$p_a=1+\frac{ab(a+b-4)}2$}
For a <projective complete intersection> curve cut out by a homogeneous <regular sequence> of positive degrees $a,b$ in $\mathbb P^3_k$, the <Koszul resolution> gives <Hilbert polynomial> $ab\,m+ab(4-a-b)/2$. Hence its <arithmetic genus> is $1+ab(a+b-4)/2$. The same resolution shows $H^0(C,\mathcal O_C)=k$, and the <dimension of a scheme> is one, so this genus equals $\dim_kH^1(C,\mathcal O_C)$. Smoothness and irreducibility of the curve are not required. For degrees five and seven, the value is $141$.
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