Tensor the sequence from part (i) with the <twisting sheaf on projective space> $\mathcal O_X(n)$:
$$
0\longrightarrow\mathcal O_X(n-d)\xrightarrow{\cdot f}\mathcal O_X(n)
\longrightarrow i_*\mathcal O_Y(n)\longrightarrow0.
$$
Its <long exact sequence in cohomology> contains
$$
H^0(X,\mathcal O_X(n))\longrightarrow H^0(Y,\mathcal O_Y(n))
\longrightarrow H^1(X,\mathcal O_X(n-d)).
$$
Since $\dim Y\geq1$, we have $r\geq2$, and the final group is intermediate cohomology of projective space. It vanishes by the <cohomology of twisting sheaves on projective space>. Hence
$$
\boxed{H^0(X,\mathcal O_X(n))\twoheadrightarrow H^0(Y,\mathcal O_Y(n))\quad\text{for every }n\in\mathbb Z.}
$$

Solution

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

 Solution

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