Let $X$ be a complex smooth projective variety, and $C\subset X$ a smooth curve. Then $C$ defines a cycle $$\beta=[C]\in H\_2(X,\mathbb Z).$$
I have a very vague question about this situation:



> 
> **Q**. If $C$ is *rigid* in $X$, how far is this condition from $C$ being the
>  unique curve on $X$ in class $\beta$?
> 
> 
> 


I would not say that rigidity implies that $C$ is the only curve in class $[C]$ (what about the converse?), but somehow I cannot quite distinguish the two situations: I have a lack of *examples* in this sense, and this is what my question is really addressing. Also: does the answer to the above question depend on $X$? What if $X$ is a Calabi-Yau threefold?


Thanks!


PS. Feel free to improve the title...


Edit. By "rigid", I mean $H^0(C,N\_{C/X})=0$, where $N\_{C/X}$ is the normal bundle. I think this is equivalent to $C$ being an isolated point in the Chow variety of $1$-cycles in $X$ (but please correct me if I am wrong).



Rigid curves, and the "richness" of their homology class
Let me spell out examples to keep in mind in both directions. I'll leave it to someone more clueful to give a better general answer.


1) Rigidity does not imply uniqueness in homology. Alex Degtyarev gave an example on ruled surfaces. Here's another. Let $C$ and $D$ be two curves in the same plane in $\mathbb P^3$, meeting transversally. Let $X$ be obtained by blowing up $C$, with exceptional divisor $E$, and let $Y$ be obtained by blowing up the strict transform of $D$ on $X$, with exceptional divisor $F$. Let $C\_0$ be the any of the fibers of $E$ that lies above one of the points of intersection of $C$ and $D$, so that $C\_0$ meets $F$. Then $C\_0$ is equivalent in homology to all of the analogous curves in fibers over the other points of intersection, of which there are $(\deg C)(\deg D)$. This is basically Hironaka's construction of a smooth non-projective threefold (after we flop $C\_0$); there's a nice picture in appendix B.3 of Hartshorne.


2) Non-rigidity does not imply non-uniqueness in homology. It's possible that $C$ is a rational curve in a threefold with $N\_{C/X} = \mathcal O \oplus \mathcal O(-2)$. Then $H^0(C,N\_{C/X}) = 1$, so it has a first-order infinitesimal deformation. However, this need not lift to an honest deformation, and so $C$ could be unique in its homology class. This situation is discussed in some detail in Reid's "Minimal models of canonical threefolds". (This is where the pictures of "Reid's pagoda" show up.)


