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> is
$$
H^i(X,\mathcal O_X(n))\longrightarrow H^i(Y,\mathcal O_Y(n))
\longrightarrow H^{i+1}(X,\mathcal O_X(n-d)).
$$
Both outer terms are intermediate cohomology groups on $\mathbb P_k^r$, because $1\leq i<i+1\leq r-1$. They vanish by the <cohomology of twisting sheaves on projective space>. Therefore
$$
\boxed{H^i(Y,\mathcal O_Y(n))=0\qquad(0<i<\dim Y,\ n\in\mathbb Z).}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-113/4/c/iv/solution