Let $X$ a degree $d$ curve in $\mathbb{P}^n$ not contained in any hyperplane and $i:X\hookrightarrow \mathbb{P}^n$ the corresponding closed immersion. Then I would like to prove that $dim H^0(X, i^{*}O_{\mathbb{P}^n}(1))=n+1$.
I think that the proof of this fact lies on the fact that it is not contained in any hyperplane so you can count linearly independent points to calculate its dimension. But I haven't understood how to deal with this kind of argument, so if you can show me how to do this (if possible) I would really appreciate it. On the other hand we have that $dimH^0(\mathbb{P}^n,O_{\mathbb{P}^n}(1))=n+1$ so I was thinking that one could exploit some cohomology argument:
If $f:X\rightarrow Y$ is an affine morphism of notherian separated schemes and $\mathcal{F}$ is a quasi-coherent sheaf of $O_X$-modules, then $H^i(X,\mathcal{F})=H^i(Y,f_*\mathcal{F})$; so I was thinking if a similar thing can be done for the pullback functor (I know that in general it doesn't hold because the pullback of an injective sheaf is not injective) and in the case it can't be done if there is a particular reason why it holds in my case.
Consider the map $\mathbb{P}^1\to\mathbb{P}^3$, given by $(t,u)\mapsto (t^d, t^{d-1}u, tu^{d-1}, u^d)$, $d\geq 3$. Easy to check that this is an embedding and if you call the image $X$, then $X$ is not contained in any hyperpalne. But $\dim H^0(X, i^*\mathcal{O}_{\mathbb{P}^3}(1))=d+1\neq 4$ if $d>3$.
The varieties $X$ with the property $\dim H^0(X,i^*\mathcal{O}_{\mathbb{P}^n}(1))=n+1$ are called linearly normal, and the example says, not all varieties are.