 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-139/1/vi/solution

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

The divisor is understood to be on

X

; the printed “on

D

h^{0} (X, O_{X} (m D)) = \frac{D ^{n}}{n !} m^{n} + O (m^{n - 1}), m \to \infty lim \frac{h ^{0} ( X , O _{X} ( m D ))}{m ^{n}} = \frac{D ^{n}}{n !} .

Since a nef divisor has nonnegative top self-intersection number, this limit is positive exactly when

D^{n} > 0

. This is the volume of a nef divisor criterion.

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