A linear family of plane cubics

The equation of a curve in this family is

          y²= x³ + x² + bx + 2b,

where  b is a parameter.  The parameter values
shown in the figure are:
 

• b = 0        

This curve  (y² = x³ + x²)  obviously has a singularity at the origin.
 
•  b = 0.05   

 
b = 0.1       
Both of these curves are nonsingular, and the real locus is connected.
 
• b = -0.05      

 
b = -0.112    
Both of these curves are nonsingular, but the real locus is disconnected.

Thus, it follows from Bertini's theorem that a general curve in this family has no singular points at finite distance. There is a base point at infinity, but the curves in the family happen to be nonsingular there.

We conclude that there are only finitely many singular curves in the family, and the parameter value  b= 0  corresponds to one of them. 

Which other parameter values correspond to singular curves ??  
 

• To answer this question, we note that a plane cubic of the form  y² = f(x)  has a singular point if and only if the polynomial  f(x)  has a multiple root. We can determine whether or not this happens by calculating the discriminant of  f(x).   For a general cubic polynomial  f(x) = x³ + ax² + bx + c,  the discriminant is given by the formula
   
          D = 4a³c - a²c² - 18abc + 4b³ + 27c².
   
  With  a = 1  and  c = 2b  this becomes  D = 4b³ + 71² + 8b = b(4b² + 71b + 8b). 
   
• So, the discrimant is = 0 for the values  b= 0,   b= -0.1134,  and  b= -17.64.
   
  • The case  b= 0  gives the curve shown in red, with a node at the origin.
     
  • The case  b= -0.1134  is not explicitly shown. It is similar to the curve shown in magenta  (b= -0.112),  but with the small oval shrunk to the single point  (x,y) = (-0.719, 0).  That point is the singular point. It is a node; however we cannot see the tangent directions because they are not real. In other words, they become "visible" only if we allow complex coordinates and work in  C²  rather than the usual real plane  R². 
     
  • The case  b= -17.64  also is not shown. As in the case  b= -0.1134  one component of the real curve is a single point  ((x,y) = (-2.781, 0)  in this case), and this is the singular point - a node with non-real tangent directions, as in the previous case. This curve and its close neighbors in the family are not shown, because the scale of the drawing would have to be larger. [Drawing to be linked at a later date.  ...]
  
  . . .
   
  [To be continued.]
   

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