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

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- Solution a

Solution

For

D = k D_{1}

, the lattice polytope of a toric divisor consists of

(m_{1}, m_{2})

satisfying

m_{1} \geq - k, m_{2} \geq 0, - m_{1} + m_{2} \geq 0, - m_{2} \geq 0.

Thus

m_{2} = 0

and

- k \leq m_{1} \leq 0

. Its lattice points index a basis of global sections, so

dim Γ (F_{1}, O (k D_{1})) = {k + 1, 0, k \geq 0, k < 0.

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