Fermat hypersurface

The degree-

d

Fermat hypersurface in

CP^{n}

is

z_{0}^{d} + \dots + z_{n}^{d} = 0.

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- Fermat hypersurface diagonal symmetry Fermat hypersurface

Fermat hypersurface diagonal symmetry

The Fermat hypersurface

z_{0}^{d} + \dots + z_{n}^{d} = 0

is preserved by multiplying each coordinate by a

d

th root of unity. Projective scalar multiplication is trivial, so the effective diagonal group is

(Z / d Z)^{n + 1} / ⟨(1, \dots, 1)⟩ ≅ (Z / d Z)^{n} .

These transformations preserve the Fubini-Study form.

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 146 / 3 / Solution

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