 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-1/25f/a/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

C

has dimension one. Its tangent has dimension one at every point except the origin and dimension two at the origin. By the smooth locus of a variety,

C_{sm} = C ∖ {(0, 0, 0)},

and the origin is its unique singular point of an algebraic variety. Each reduced point component has dimension zero and zero-dimensional tangent space, so both

P

and

Q

are smooth on their respective components.

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