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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 24F / c / Solution Created 2026-09-24 Updated 2026-09-29

Suppose an affine line lies in

W

. Parametrize it as

(x, y, z) = (a_{1}, a_{2}, a_{3}) + t (b_{1}, b_{2}, b_{3}) .

The defining equation would give the polynomial identity

(a_{1} + t b_{1}) (a_{2} + t b_{2}) (a_{3} + t b_{3}) = 1 in k [t] .

The only units of

k [t]

are nonzero constants. Since a product is a unit only when every factor is a unit, all three linear polynomials must be constant, so

b_{1} = b_{2} = b_{3} = 0

. The alleged line is a point, a contradiction. Therefore

W = V (x yz - 1) contains no affine lines .

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