Arithmetic adjunction formula on a smooth surface ($2p_a(C)-2=C^2+K_X\cdot C$)

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 $0\to {\mathcal{O}}_X(-C)\to {\mathcal{O}}_X\to {\mathcal{O}}_C\to 0$ to derive it. This arithmetic version applies also to singular curves and in arbitrary characteristic.