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

Table of contents
- a ii
  - Solution a
- b ii
  - Solution b

a

Solution

For every integer

t

,

P (P_{k}^{2}, O, t) = χ (P_{k}^{2}, O (t)) = (2 t + 2) = \frac{t ^{2} + 3 t + 2}{2} .

For

t \geq 0

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.

b

Solution

A smooth plane curve of degree

m

has genus

g = \frac{( m - 1 ) ( m - 2 )}{2} .

The line bundle

O_{X} (D) (t)

has degree

d + m t

. The Riemann-Roch theorem therefore gives

P (X, O_{X} (D), t) = χ (O_{X} (D) (t)) = m t + d + 1 - g = m t + d + 1 - \frac{( m - 1 ) ( m - 2 )}{2} .

Solved by gpt-5.6-sol high.

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