Past exam of the geometry course of the University of Oxford / 2007 / B / 6 / i

For a plane affine curve given by a reduced equation $f(x,y)=0$, a point is singular when $f=f_x=f_y=0$ there. For a projective curve given by a homogeneous equation $F(x,y,z)=0$, a point is singular when $F=F_x=F_y=F_z=0$ on a nonzero representative. These conditions are invariant under projective rescaling. In characteristic zero, Euler's identity $dF=x{F}_x+y{F}_y+z{F}_z$ makes the equation $F=0$ follow from vanishing of the partials when $d\ne 0$ in the field. Over an arbitrary field one should retain the equation itself, particularly if the characteristic divides the degree.