global sections of some sheaves Let $Y$ be a connected, nonsingular, positive dimensional subvariety of $\mathbb{P}^n_k$ over an algebraically closed field $k$, and let $\mathcal{I}$ be the ideal sheaf of $Y$.
Why are the followings true?
1) $\Gamma(Y,\mathcal{I}^r/\mathcal{I}^{r+1})=0$ for any $r\ge 1$.
2) $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^r)=k$ for any $r\ge 1$.
Could someone explains this for me, thanks.
 A: Write for simplicity $X=\mathbb{P}^n$. 
The easiest way of showing 1) is probably by noting that $\mathcal I / \mathcal I^2$ injects as a subbundle of $\mathcal O_Y(-1)^{n+1}$ (this follows from combining the conormal sequence with the Euler sequence) and so none of it's symmetric powers $S^r(\mathcal I / \mathcal I^2)=\mathcal I^r / \mathcal I^{r+1}$ can have any global sections. Now taking the cohomology sequence of
$$
0 \to \mathcal{I}^r/\mathcal{I}^{r+1}\to\mathcal{O}_X/\mathcal{I}^{r+1} \to\mathcal{O}_X/\mathcal{I}^{r}  \to 0,
$$shows that $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r+1})$ injects into $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r})$, and so by induction on $r$, we get 2).
A: Put $\mathcal N=(\mathcal I/\mathcal I^2)^\vee$, the normal bundle to $Y$ in $X=\mathbb P^n$.
The tangent bundle sequence $0\to T_Y\to (T_X)\vert_Y\to \mathcal N\to 0$ and the ampleness of $T_X$ shows that $\mathcal N$ is ample. Now $\mathcal I^r/\mathcal I^{r+1}$ is the dual of the $r$th symmetric power of $\mathcal N$, so cannot have sections.
