A combinatorial description of finite O-sequences and aCM genera

The goal of this paper is to explicitly detect all the arithmetic genera of arithmetically Cohen–Macaulay projective curves with a given degree d. It is well-known that the arithmetic genus g of a curve C can be easily deduced from the h-vector of the curve; in the case where C is arithmetically Cohen–Macaulay of degree d, g   must belong to the range of integers {0,…,(d−12)}. We develop an algorithmic procedure that allows one to avoid constructing most of the possible h-vectors of C. The essential tools are a combinatorial description of the finite O-sequences of multiplicity d, and a sort of continuity result regarding the generation of the genera. The efficiency of our method is supported by computational evidence. As a consequence, we single out the minimal possible Castelnuovo–Mumford regularity of a curve with Cohen–Macaulay postulation and given degree and genus.