= Top-twist vanishing by hyperplane induction
{title2=$H^n(\mathbf P^n,\mathcal O(m))=0\quad(m\ge-n,\ n>0)$}
The <hyperplane exact sequence for twisting sheaves> identifies consecutive top-degree groups whenever the hyperplane's preceding-degree group vanishes. Induction on dimension, starting with the surjective restriction of constants to a point on $\mathbf P^1$, propagates the degree-zero top vanishing down to $m=-n$. Its top-degree surjections also propagate vanishing to every positive twist. This proof does not apply the negative-twist global-section formula to $\mathbf P^0$.
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