For every integer $t$,
$$
P(\mathbb P_k^2,\mathcal O,t)=\chi(\mathbb P_k^2,\mathcal O(t))
=\binom{t+2}{2}
=\frac{t^2+3t+2}{2}.
$$
For $t\geq0$ this is the dimension of the homogeneous polynomials of degree $t$, since the higher cohomology vanishes; polynomiality then identifies the <Hilbert polynomial>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 126 / 2 / ii / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-126/2/ii/a/solution