Riemann–Roch theorem for algebraic surfaces ($\chi ({\mathcal{O}}_X(D))=\chi ({\mathcal{O}}_X)+\frac{1}{2}D\cdot (D-K_X)$)

For a Cartier divisor on a smooth projective surface, the displayed formula computes its Euler characteristic of a coherent sheaf. Combined with Serre duality, it turns intersection numbers into lower bounds for section spaces.