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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-3/24f/iii/solution

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 24F / iii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

The intersection with a line

ℓ

is a hyperplane section:

D = ℓ \cap C, O_{C} (D) ≅ O_{C} (1) .

(1)

The canonical bundle of a smooth projective hypersurface formula, obtained from the Adjunction formula, gives for a smooth degree-four curve in

P^{2}

K_{C} ≅ (K_{P^{2}} \otimes O_{P^{2}} (4)) ∣_{C} ≅ O_{C} (- 3 + 4) ≅ O_{C} (1) .

(2)

Thus $O_{C} (D) ≅ K_{C}$ , so the divisor class of $D$ is the canonical divisor class. Therefore $D$ is a canonical divisor. Its degree is also $4 = 2 g - 2$ , consistently with $g = 3$ .

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