The title says it all, what's a good dense open of $\bar{M}\_g,n(X,\beta)$ which play the role of ${M}\_g$ in $\bar{M}\_g$?


My first (naive) guess is maps from a genus $g$ smooth curve to $X$ which represents the class $\beta$. But I'm a little bit concerned, is it dense for sure? Could it happen that in some cases one has singular curves only? Can a stable map from a singular curve always deform to a map from a smooth curve of genus $g$?



What's a good dense open of $\bar{M}_g,n(X,\beta)$?
If $X$ is convex and $g = 0$, then you can take smooth curves with distinct marked point, this will be dense. However, in general the locus of smooth curves is not dense. An easy example is that of degree 1 unpointed maps from a curve of genus $g > 0$ to $\mathbb P^1$; such a curve must consist of a copy of $\mathbb P^1$ mapping isomorphically and some vertical components, hence it can not be smooth.


In general I would guess that there is no analogue of $M\_g$. There are good philosophical reasons why it has to be so; the space $\overline{M}\_g,n(X,\beta)$ is much too big, in general, and badly singular, it can't be expected to behave nicely in any way. The “good” part of $\overline{M}\_g,n(X,\beta)$ is the virtual fundamental class; but that's just a class of cycles, it does not have open parts.


