Solution
= Solution
The curve $C$ 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>,
$$
\boxed{C_{\mathrm{sm}}=C\setminus\{(0,0,0)\}},
$$
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.