= Intermediate cohomology vanishing for a projective complete intersection
For a length-$r$ <projective complete intersection> $X_r\subseteq\mathbb P^N_k$, the groups $H^q(X_r,\mathcal O_{X_r}(m))$ vanish for every integer $m$ and $0<q<N-r$. Starting with <cohomology of twisting sheaves on projective space>, induct through $0\to\mathcal O_{X_{j-1}}(m-d_j)\to\mathcal O_{X_{j-1}}(m)\to\mathcal O_{X_j}(m)\to0$. Both adjacent groups in the <long exact sequence in sheaf cohomology> vanish precisely in the indicated range.
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