Smoothness of the images of members of a linear system under an endomorphism of the affine plane

Let φ=(f,g)φ=(f,g) be an endomorphism of the affine plane C2C2 defined by two polynomials f,g∈C[x,y]f,g∈C[x,y] and let Λ={Cb∣b∈C}Λ={Cb∣b∈C} be the pencil of lines CbCb defined by x=bx=b. We shall consider the smoothness criterion of the image curve φ(Cb)φ(Cb). The hypersurface VV whose coordinate ring is C[x,f,g]C[x,f,g] and the normalization V˜ of VV will play interesting roles in analyzing the properties of the set φ(Λ)={φ(Cb)∣b∈C}φ(Λ)={φ(Cb)∣b∈C}.