- Follows since $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^r)$ is the global sections of the structure sheaf of the scheme $rY$ defined by $\mathcal{I}^r$ (which is connected). For 1), take the long exact sequence of cohomology of $$ 0 \to \mathcal{I}^r/\mathcal{I}^{r+1}\to\mathcal{O}_X/\mathcal{I}^{r+1} \to\mathcal{O}_X/\mathcal{I}^{r} \to 0. $$
J.C. Ottem
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