Determinantal variety

A determinantal variety is cut out by minors imposing an upper bound on matrix rank.

The projective locus wherehas rank one is cut out by its three $2\times 2$ minors. It is a smooth surface: on each of the charts $y_0\ne 0$, $y_1\ne 0$, $y_3\ne 0$, and $y_4\ne 0$, the equations eliminate three coordinates and leave two free affine coordinates; these charts cover the locus.

The variety of $m\times n$ matrices of rank at most one has dimension $m+n-1$. For $m,n\ge 2$, its only singular point is the zero matrix.