Genus of an embedded curve in projective smooth manifold deformed in its homology class Let $C$ be a smooth embedded curve in an $n$-dimensional complex projective smooth manifold $X$ of class $[C]=\beta \in H_2(X,\mathbb{Z}).$ 

  
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*Can one make arbitrary arithmetic genus by deforming $C$ while keeping $\beta$ fixed (i.e. within its class $\beta$)? if so, then is it possible to explicitly
  construct such deformations? 
  
*The same question but with the only difference on keeping $\beta$ and the smoothness structure both fixed?

For the first part of 1), I think, one can add mild singularities to $C$ while preserving $\beta$ to make arbitrary genus, but I don't know how to construct such an example. 
 A: I guess I am more than three years late, but here is a thought.
One must be clear about what it means to "fix the genus" of your curves, and what is meant by "curve".
My guess is that you are looking at the Hilbert scheme
$$H_X(\beta,r)=\{Z\subset X\,:\,\chi(\mathscr O_Z)=r,\,[Z]=\beta\}.$$
Here you have different kinds of objects:


*

*smooth pure curves;

*smooth, or just Cohen-Macaulay, curves with some points (embedded or isolated);

*worse stuff.


These three items deform into one another when moving in flat families. 
Fix your smooth genus $g$ curve $C$ embedded in the complex manifold $X$. A way to keep $\beta$ fixed and change arithmetic genus is to add floating points (embedded or isolated, reduced or fat). If $C\subset Z\subset X$ is the resulting subscheme, then you suddenly have $\chi(\mathscr O_Z)=\chi(\mathscr O_C)+m$, where $m$ is the number of points you have added. Clearly, $[Z]=[C]=\beta$ stays unchanged. But notice that $C$ and $Z$ now belong to different Hilbert schemes:
$$C\in H_X(\beta,1-g),\qquad Z\in H_X(\beta,\chi(\mathscr O_C)+m).$$
If you aim at staying in the same Hilbert scheme, then $\chi(\mathscr O_Z)$ must not change. What is true is that in a flat family, $\chi(\mathscr O_Z)$ stays constant (so, strictly speaking the arithmetic genus $1-\chi(\mathscr O_Z)$ also stays constant!), but the arithmetic genus of the Cohen-Macaulay supports (the "actual" curves) of the subschemes can change: when it increases, embedded points appear (see the example below). 
In general, for a point $Z\in H_X(\beta,r)$, you have the following. Let $C\subset Z$ be the maximal Cohen-Macaulay subscheme of $Z$, then $$r=1-p_a(C)+\chi(\mathscr I_C/\mathscr I_Z),$$ where $\chi(\mathscr I_C/\mathscr I_Z)$ is the "number of points".
Example. In Hartshorne's book, there is at some point the example of a family of twisted cubics (genus zero) degenerating to a plane nodal cubic (genus one) with an embedded point. It is a flat family of subschemes of $\mathbb P^3$ of arithmetic genus zero. However you can also decide to smooth out the node and let the embedded point roam all over $\mathbb P^3$ freely. So now you have a new flat family where you still have arithmetic genus zero everywhere, and the "actual" curves are smooth with different genera: almost all of them are smooth of genus zero, and one is smooth of genus one (this has a free point to save flatness!).
