Consider the Hilbert scheme parametrizing the curves of degree $d$ and arithmetic genus $g$ in $\mathbf{P}^n$. Is there a formula for its dimension in terms of $g,n,d$? Is there a bound on its number of irreducible components?

It is well known that if $C \subset \mathbf{P}^n$ has degree $d$, arithmetic genus $g$ and is a locally complete intersection, then the dimension at $C$ of the Hilbert scheme $\mathscr{H}=\mathscr{H}^n_{f(t)}$, with $f(t)=dtg+1$, satisfies $$\dim _C \mathscr{H} \geq h^0(C, \mathscr{N}_C)h^1(C, \mathscr{N}_C), \quad (*)$$ where $\mathscr{N}_C$ is the normal sheaf of $C$ in $\mathbf{P}^n$. When $C$ is a smooth and irreducible, by using RiemannRoch one checks that the right hand side of $(*)$ equals $p(n,d,g):=(n+1)d+ (n3)(1g).$ A component of $\mathscr{H}$ of dimension exactly $p(n,d,g)$ is called regular, whereas a component of dimension strictly bigger that $p(n,d,g)$ is called superabundant. For instance, it is known that every complete intersection curve $C$ belongs to a regular component of $\mathscr{H}$. 

