Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 113 / 3 / d / Solution

Multiplication by the homogeneous equation $f_d$ and restriction to its zero scheme give the structure-sheaf sequence of a hypersurfaceBy cohomology under a closed immersion, $H^q({\mathbb{P}}^3,i_{\ast }{\mathcal{O}}_X)\cong H^q(X,{\mathcal{O}}_X)$. The associated long exact sequence in cohomology containsBoth outer groups vanish by the cohomology of twisting sheaves on projective space, because they are intermediate cohomology groups on ${\mathbb{P}}^3$. Therefore