I want to ask about the Jacobi criterion for checking smoothness of this projective variety (i'll write the coordinates as $x,y,z,w$), I need to find singular points of:
$$ xyz + xyw +xzw+ yzw=0 $$
I think that I can't parametrize this variety so I can't check if it is smooth using the fact that maybe is isomorphic to something easyier. But I think that use jacobi criterion is really long ...so how can I proceed? It exists a smarter way?