• Introduction
• Singularities
• Duality map
• Dual parametrization
• Dual equation
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A parametrization of the dual surface:
 
    The approach used here also is based on what I have learned from Sevín Recillas.  We'll study both the parametric equations of the dual surface and the implicit equation of the dual surface.

    We start with a version of the implicit equation of the Cayley surface which is valid at points of the Cayley surface where all four coordinates are nonzero.  The idea is to divide the implict equation by  wxyz,  so that we work with the implicit equation  G(w,x,y,z) = 0,  where  G(w,x,y,z) = ¹⁄_w + ¹⁄_x + ¹⁄_y + ¹⁄_z.  This leads to the following version of the duality map:
 
      (w:x:y:z) ---> (^∂G⁄_∂w: ^∂G⁄_∂x: ^∂G⁄_∂y: ^∂G⁄_∂z) = (¹⁄_w² : ¹⁄_x² : ¹⁄_y² : ¹⁄_z² ).
 
(A minus sign has been cancelled in the last homogeneous coordinate vector.) 

    We use a similar approach to study the parametric equations.  Given a point  (w:x:y:z)  of the Cayley surface, with  wxyz ≠ 0,  we set  s = ¹⁄_x,  t = ¹⁄_y,  and  u = ¹⁄_z.  We use the implicit equation to show that  ¹⁄_w = s+t+u.   Therefore, we have the following parametrization of the dual surface:
 
      (s:t:u) ---> ((s + t + u)²: s²: t²: u²).
 
Since there are no denominators and no common factors, this gives a globally defined parametrization of the dual surface,  mapping  P²  onto the dual surface in  P³. 

    The parametrization clearly is finite-to-one.  By lightly modifying the methods that were used in the discussion of the duality map, we can show that:

• The parametrization is one-to-one except at points (in  P²)  of the 3 lines  s=-t;  s=-u;  and  t=-u. 
   
• The parametrization is generically two-to-one along each of these lines.  For instance, at points of the line  s=-t  the map is given by:  
        (s:-s:u) ---> (u²: s²: s²: u²),  
  and this line is mapped onto the line  w=z,  x=y  in  P³,  so that this line necessarily lies on the dual surface.  The restriction to the line  s=-t  thus has two ramification points, namely  (0:0:1)  and  (1:-1:0).  The images of these ramification points turn out to be pinch points of the dual surface.  {Similarly, our other two lines in  P²  map to two other lines which lie on the dual surface.  Each of these lines contains two pinch points of the dual surface.} 
   
• Each of the points  (1:1:-1),  (1:-1:1)  and  (-1:1:1)  of  P²  is mapped to the point  (1:1:1:1)  of the dual surface.  All three of the lines that lie on the dual surface contain this point, and it is a triple point of the dual surface. 

    One other interesting property of the dual surface can be easily seen from this parametrization.  Specifically, it is that four lines in  P²  are mapped to plane conic curves on the dual surface. 

• For instance, the line  u = 0  is mapped as follows:  
        (s:t:0) ---> ((s + t)²: s²: t²:0).
  The image obviously is a curve of degree 2 in the plane  z = 0;  its implicit equation is  (w-x-y)² = 4xy. 
   
• Similarly, the lines  t = 0  and  u = 0  are mapped to curves of degree 2 in the planes  y = 0  and  x = 0  respectively. 
   
• Finally, the line  s+t+u = 0  is mapped as follows: 
        (-t - u :t :u) ---> (0:(t + u)²:t²: u²).
  In this case, the image is a curve of degree 2 in the plane  w = 0,  and the implicit equation of the image curve is  (x-y-z)² = 4yz.