Serre and the Riemann–Roch Problem

Since the beginnings of the development of the theory of algebraic surfaces, geometers have tried to prove a generalization of the Riemann–Roch theorem for algebraic curves (see Zariski’s account in [195, Chap. IV], as well as Gray ’s article [84]). Max Noether formulated such a generalization in his 1886 paper [144]. But he could only prove an inequality (I use the notations and the explanations from [32, Chaps. 7 and 35])