Veronese Surface

A smooth 2-D surface given by embedding the Projective Plane into projective 5-space by the homogeneous parametric equations

$$v(x,y,z)=(x^2,y^2,z^2,xy,xz,yz).$$

The surface can be projected smoothly into 4-space, but all 3-D projections have singularities (Coffman). The projections of these surfaces in 3-D are called Steiner Surfaces. The Volume of the Veronese surface is $2\pi^2$.

See also Steiner Surface

References

Coffman, A. ``Steiner Surfaces.'' http://www.ipfw.edu/math/Coffman/steinersurface.html.