Let $C\subset X$ be a smooth irreducible curve of genus $g$, embedded in a smooth projective 3-fold $X$. So its homology class $\beta=[C]\in H_2(X)$ is an irreducible class. I want to compare two moduli schemes:

- The Chow scheme $CH_1(X,\beta)$ of 1-cycles of degree $\beta$ on $X$, and
- the Hilbert scheme $I^{CM}_{1-g}(X,\beta)$ of Cohen-Macaulay curves of arithmetic genus $g$ in class $\beta$.

The latter is an open subscheme of the full Hilbert scheme $I_{1-g}(X,\beta)$, parametrizing subschemes $Z\subset X$ such that $\chi(\mathscr O_Z)=1-g$ and $[Z]=\beta$. I do not know much about the former.

The notation for the Chow scheme does not include the genus, according to the fact that the arithmetic genus of a CM curve in class $\beta$ can change -- even for irreducible classes, I think. I cannot figure any (set-theoretic) difference between the Chow scheme and the space of *all* Cohen-Macaulay curves in class $\beta$ (although I never heard of a proper definition of the latter). So here is my question:

Are the moduli schemes $I^{CM}_{1-g}(X,\beta)$, with varying $g$, exactly the connected components of $CH_1(X,\beta)$? Are they at least open in the Chow scheme?

A comment: how many $g$? According to this post, the arithmetic genus of a CM curve is only bounded from **above** for fixed $\beta$, but since our $\beta$ is irreducible, I think all curves in class $\beta$ will be reduced, so we should get a lower bound as well.

**Edit**. By *irreducible* curve class, I mean that one cannot write $\beta=\beta_1+\beta_2$, with $\beta_i$ the class of a curve.

Thanks!