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

Tensor the sequence from part (i) with the twisting sheaf on projective space ${\mathcal{O}}_X(n)$:Its long exact sequence in cohomology containsSince $\dim Y\ge 1$, we have $r\ge 2$, and the final group is intermediate cohomology of projective space. It vanishes by the cohomology of twisting sheaves on projective space. Hence