It is well known that if $C \subset \mathbf{P}^n$ has degree $d$ and arithmetic genus $g$, then the dimension at $C$ of the Hilbert scheme $\mathscr{H}$ satisfies $$\dim _C \mathscr{H} \geq h^0(C, \mathscr{N})-h^1(C, \mathscr{N}), \quad (*)$$
where $\mathscr{N}$ is the normal sheaf of $C$ in $\mathbf{P}^n$.

When $C$ is smooth and irreducible, by using Riemann-Roch one checks that the right hand side of $(*)$ equals $p(n,d,g):=(n+1)d+ (n-3)(1-g).$ 

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*.