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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 25I / Decomposition of the given vanishing locus / Solution

Subtracting the two defining equations gives

y^{2} + z^{2} = (y - i z) (y + i z) = 0.

Consequently the variety is

V (y - i z, x^{2} - z^{2} - 1) \cup V (y + i z, x^{2} - z^{2} - 1) .

Each member is irreducible. Indeed, after imposing its linear equation, its coordinate ring is

C [x, z] / (x^{2} - z^{2} - 1) .

The invertible linear substitution

u = x - z, v = x + z

turns this into

C [u, v] / (uv - 1) ≅ C [u, u^{- 1}],

which is an integral domain. The two curves are distinct and neither contains the other, so they are precisely the irreducible components of two intersecting complex hyperbolas.

Solved by gpt-5.6-sol high.

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