Smooth projective surface from overlapping rank-one coordinates
= Smooth projective surface from overlapping rank-one coordinates
The projective locus where
$$
\begin{pmatrix}y_0&y_1&y_2\\y_2&y_3&y_4\end{pmatrix}
$$
has rank one is cut out by its three $2\times2$ minors. It is a smooth surface: on each of the charts $y_0\ne0$, $y_1\ne0$, $y_3\ne0$, and $y_4\ne0$, the equations eliminate three coordinates and leave two free affine coordinates; these charts cover the locus.