Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 113 / 4 / b

The cohomology of twisting sheaves on projective space isIn particular, $H^i(X,{\mathcal{O}}_X)=H^i(X,{\mathcal{O}}_X(1))=0$ for $i>0$, while $H^0(X,{\mathcal{O}}_X)=k$ and $H^0(X,{\mathcal{O}}_X(1))\cong k^{r+1}$.

Apply the long exact sequence in cohomology to the displayed Euler sequence. Its degree-zero part isand all later terms vanish. The first map sends $1$ to the tuple of homogeneous coordinates and is injective. Therefore