Let $X$ be a smooth projective variety. Given fixed Chern classes with positive rank $r$, let $M$ be the moduli space of semistable sheaves with such Chern classes. In general we only know $M$ is a projective scheme. (if $\mathrm{dim}(X)$ is $1,2$ a lot more can be said)

1st Question: Is there an example where $M$ is smooth but not connected?

2nd Question: I am trying to show the following scenario cannot happen: $M$ is smooth of dimension $d$ but disconnected and one of the connected components consists merely of stable vector bundles. I feel we might be able to use the fact that the component only contains vector bundles to contradicts the projectivity but not sure how.

Thanks in advance!

Let $X$ be a smooth projective variety. Given fixed Chern classes with positive rank $r$, let $M$ be the moduli space of semistable sheaves with such Chern classes. In general we only know $M$ is a projective scheme. (if $\mathrm{dim}(X)$ is $1,2$ a lot more can be said)

1st Question: Is there an example where $M$ is smooth but not connected?

2nd Question: I am trying to show the following scenario cannot happen: $M$ is smooth of dimension $d\geq 1$(Edit: first version did not contain $\geq 1$, see Jason Starr's comments below for an interesting example for $d=0$) but disconnected and one of the connected components consists merely of stable vector bundles. I feel we might be able to use the fact that the component only contains vector bundles to contradicts the projectivity but not sure how.

Thanks in advance!