  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
$\big\{0,\ldots,\binom{d-1}{2}\big\}$. 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.
