Arithmetic adjunction formula on a smooth surface
= Arithmetic adjunction formula on a smooth surface
{title2=$2p_a(C)-2=C^2+K_X\cdot C$}
For an integral curve $C$ on a <smooth projective surface>, $2p_a(C)-2=C\cdot(C+K_X)$. Apply <Riemann–Roch theorem for algebraic surfaces> to the <divisor restriction exact sequence> $0\to\mathcal O_X(-C)\to\mathcal O_X\to\mathcal O_C\to0$ to derive it. This arithmetic version applies also to singular curves and in arbitrary characteristic.