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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-25/6/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 25 / 6 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

The adjunction formula on the degree-

d

projective hypersurface gives

K_{X} = (d - 4) H

. Since

C

is a smooth rational curve, adjunction on

C

gives

(K_{X} + C) \cdot C = 2 g (C) - 2 = - 2

. Its degree is

H \cdot C = a

, so the smooth rational-curve self-intersection on a projective hypersurface formula is

C^{2} = - 2 - (d - 4) a .

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