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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-139/2/iv/solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 139 / 2 / iv / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

A connected integral projective variety has only constant regular functions, so

h^{0} (X, O_{X}) = 1

. By Serre duality and

K_{X} \sim 0

,

h^{2} (X, O_{X}) = h^{0} (X, O_{X} (K_{X})) = 1

. The assumed first sheaf cohomology vanishes. Therefore the Euler characteristic of a coherent sheaf is

χ (X, O_{X}) = 1 - 0 + 1 = 2.

These hypotheses characterize the K3 surface setting used below.

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