EDIT: As Angelo mentions, the argument below has a problem in the case of non-reduced curves.  I am not sure that an upper bound for the arithmetic genus is impossible to show, but certainly the lower bound for Cohen-Macaulay curves is false, as Angelo's examples show.  The argument below shows that there is an upper bound (and in fact there is also a lower bound) for the arithmetic genus of a reduced subscheme of pure dimension one in projective space, whether Cohen-Macaulay or not.  I tried to play a little with the Cohen-Macaulay condition to prove that there is an upper bound, but with little success.

Choose an embedding of *X* in projective space. Since your curves are all in the same homology class, they all have the same degree: this is simply the intersection number of the ample class with the homology class of the curves. Generic projection to a plane tells you (since the curves are **reduced**) that the curves you are interested in are partial normalizations of plane curves with bounded degree. Since the arithmetic genus decreases under (partial) normalizations, and since the arithmetic genus of a plane curve of bounded degree is bounded **above**, you conclude that the arithmetic genera of your curves are bounded **above**.