For $D=kD_1$, the <lattice polytope of a toric divisor> consists of $(m_1,m_2)$ satisfying
$$
m_1\geq-k,\qquad m_2\geq0,\qquad-m_1+m_2\geq0,\qquad-m_2\geq0.
$$
Thus $m_2=0$ and $-k\leq m_1\leq0$. Its lattice points index a basis of global sections, so
$$
\dim\Gamma(\mathbb F_1,\mathcal O(kD_1))=
\begin{cases}
k+1,&k\geq0,\\
0,&k<0.
\end{cases}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 152 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-152/3/a/solution