Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 113 / 4 / c / iv / Solution

Use again the twisted ideal-sheaf sequence of a projective hypersurface. For $0<i<\dim Y=r-1$, the relevant part of its long exact sequence in cohomology isBoth outer terms are intermediate cohomology groups on ${\mathbb{P}}_k^r$, because $1\le i<i+1\le r-1$. They vanish by the cohomology of twisting sheaves on projective space. Therefore