The <cohomology of twisting sheaves on projective space> is
$$
H^p(\mathbb P_k^r,\mathcal O(n))\cong
\begin{cases}
k[x_0,\ldots,x_r]_n,&p=0,\ n\geq0,\\
k[x_0,\ldots,x_r]_{-n-r-1}^{*},&p=r,\ n\leq-r-1,\\
0,&\text{otherwise}.
\end{cases}
$$
In 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 is
$$
0\longrightarrow k\longrightarrow (k^{r+1})^{\oplus(r+1)}
\longrightarrow H^0(X,\mathcal T)\longrightarrow0,
$$
and all later terms vanish. The first map sends $1$ to the tuple of homogeneous coordinates and is injective. Therefore
$$
\boxed{H^i(X,\mathcal T)\cong
\begin{cases}
k^{(r+1)^2-1},&i=0,\\
0,&i>0.
\end{cases}}
$$

Solution

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

 Solution

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