= Nonnegative hyperplane twist cohomology by restriction
{title2=$h^0(\mathbf P^n,\mathcal O(m))=\binom{n+m}{n},\quad H^{r>0}=0$}
For $m\ge0$, these <sheaf cohomology> dimensions follow from the vanishing for $\mathcal O$ and the <hyperplane exact sequence for twisting sheaves>. Induct on dimension and then on $m$ in $0\to\mathcal O(m-1)\to\mathcal O(m)\to i_*\mathcal O_{\mathbf P^{n-1}}(m)\to0$. The <long exact sequence in sheaf cohomology> gives higher-degree vanishing and the recurrence $h^0_n(m)-h^0_n(m-1)=\binom{n-1+m}{n-1}$. Starting with $h^0_n(0)=1$ yields the formula; the corresponding homogeneous monomials form a basis of global sections.
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