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Irreducible components of two intersecting complex hyperbolas

The algebraic set

V (x^{2} + y^{2} - 1, x^{2} - z^{2} - 1) \subseteq A_{C}^{3}

has the two irreducible components

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

Each is irreducible because the linear change

u = x - z

,

v = x + z

identifies its coordinate ring with

C [u, v] / (uv - 1) ≅ C [u, u^{- 1}],

an integral domain.

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Solution