I'm having trouble to find a theorem about this:
Let's $F_z$ be a family of curves defined by $$F_z : F(x,y) + z = 0$$ where $F$ is an irreducible polynomial and $z \in \mathbb{C}$. My question is: is it true that the set of values of $z$ for which $F_z$ is singular, is finite?. As far as I know all I can say is that the set of smooth curves is dense.
The condition that the curve $F(x,y)+z=0$ be smooth is that it does not share any points of intersection with the curves $(\partial F/\partial x) (x,y)=0$ and $(\partial F/\partial y) (x,y)=0$. Since $F$ is irreducible, the intersection of these three curves consists of finitely many points $(x_i, y_i)$, by Bezout's theorem. Thus $F(x,y)+z=0$ will be smooth whenever $z$ is different from one of the finitely many values $-F(x_i, y_i)$.