I'm trying to solve Exercise 1.26 from the book "Moduli of Curves" 
by Harris and Morrison on page 14:

***Exercise (1.26)*** Determine the normal bundle to the rational normal
curve $C \subset \mathbb{P}^r =: X$ and show, by computing its $h^0$, 
that the Hilbert scheme
parameterizing such curves is smooth at any point corresponding to
a rational normal curve.

The part I can't solve is that one how the dimension of global sections
of the normal bundle $h^0= \dim_{\mathbb{C}}H^0(C, N_{C / X})$  imply that the Hilbert scheme $\mathcal{H}$ 
is smooth at $[C]$.



Since $C$ is rational normal curve $C \cong \mathbb{P}^1$ and the long
exact cohomology sequence of the
exact sequence $0\rightarrow \mathcal{T}_{C} \rightarrow 
\mathcal{T}_{X}\otimes\mathcal{O}_{C} \rightarrow 
\mathcal{N}_{X|C} \rightarrow 0$ gives

$$\operatorname{dim} H^0 (C,N_{C/X}) = \operatorname{dim} 
H^0 (C,T_X \otimes O_C) -3 = (r+1) \cdot \dim \ H^0 (C,O_C(1))-3$$

(the last one is consequence of Euler sequence). Fine now we know $h^0$. Immediately before it was shown that the tangent space of 
$\mathcal{H}$ at $[C]$ is

$$T_{[C]}\mathcal{H}= H^0(C, N_{C / X})$$

Per definition a scheme $X$ is smooth at a point $P$ if the dimension of 
tangent space at this point coinsides with *local dimension*: ie there exist
an open affine subscheme $U \subset X$ with $P \in U$ and 
$\dim_P T= \dim \ U$.

Working through previous pages I nowhere found informations about 
local dimension of the Hilbert scheme parametrizing rational normal
curve so I have no idea how can I compare the dimension of 
$T_{[C]}\mathcal{H}= H^0(C, N_{C / X})$ which I calculated above
 with the local dimension of $\mathcal{H}$.

Can anybody give me some hints how to attack this part of the exercise?