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

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

(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?

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?

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 $0\rightarrow \mathcal{T}_{C} \rightarrow \mathcal{T}_{X}\otimes\mathcal{O}_{C} \rightarrow \mathcal{N}_{C|X} \rightarrow 0$ gives

$$\operatorname{dim} H^0 (Y,N) = \operatorname{dim} H^0 (C,T_X \otimes O_Y) -3 = (r+1) \cdot \dim \ H^0 (C,O_C(1))-3$$

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

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

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

(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?