A linear family of plane cubics

This example is a linear family of plane cubic curves, which has no base points at finite distance.

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:

The defining equation of the family is obviously linear in the parameter b, and it is easily checked that there are no base points at finite distance.

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 ??  

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