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,…,(2d−1)}. 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.Comment: Final versio