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.