A bound on the plurigenera of projective surfaces

We exhibit a sharp Castelnuovo bound for the ith plurigenus of a smooth minimal surface of general type and of given degree d in the projective space , and classify the surfaces attaining the bound, at least when d⪢r. We give similar results for surfaces not necessarily minimal or of general type, but only for i⪢r (however, in the case r≤8, we give a complete classification, i.e., for any i≥1). In certain cases (only for r≥12) the surfaces with maximal plurigenus are not Castelnuovo surfaces, i.e., surfaces with maximal geometric genus.