Genus of an embedded curve in projective smooth manifold deformed in its homology class

Question

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

1. 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?

2. 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.