Number of singular fibers in families of hypersurfaces Consider the projection map 
$$\pi: X = V(t_0 f + t_1 gh) \to \mathbf P^1,$$
where $[t_0: t_1]$ are the homogeneous coordinates on $\mathbf P^1$, $f=f(x_0, \dots, x_n)$ is a homogeneous polynomial of some degree $d$ such that $V(f) \subset \mathbf P^n$ is smooth and $g$ and $h$ satisfy $\deg g+\deg h = d$ and they similarly define smooth hypersurfaces in $\mathbf P^n$ such that $V(gh) \subset X$ is a normal crossings divisor, i.e., $V(g) \cap V(h)$ is again smooth of the expected dimension.
As the fiber $V(f) \cong \pi^{-1}([1:0])$ is smooth by assumption, the map is generically smooth, so has only a finite number of singular fibers. 

Question: Is there an upper bound on the number of singular fibers of
  $\pi$? (In terms of the degree of $f$, for example?)

(If there is an answer to a similar question for more general families of hypersurfaces, I am interested in this, too.)
Thank you!
 A: Definitely there is an upper bound for the degree, namely the degree $(n+1)(d-1)^{n-1}$ of the discriminant hypersurface $\Delta$ in the projective space $\mathbb{P} S_d$ associated to the finite dimensional $k$-vector space $S_d = k[x_0,\dots,x_n]_d$ of degree $d$ homogeneous polynomials in $x_0,\dots,x_n$.  If you look at my previous answer (or look this up in Fulton's "Intersection Theory", Eisenbud-Harris's "3264 and All That" or various other sources), you will see that the degree of the discriminant hypersurface is computable (in a more general context).  Here it is particularly simple, just $(n+1)(d-1)^{n-1}$.  
The point is, your pencil of hypersurfaces gives a morphism $f:\mathbb{P}^1 \to \mathbb{P} S_d$ that is an isomorphism onto a line $\Lambda$ in $\mathbb{P} S_d$.  So now you are asking the cardinality of the intersection $\Lambda \cap \Delta$, under your hypothesis that $\Lambda$ is not contained in $\Delta$.  As $\Delta$ is just a hypersurface and $\Lambda$ is just a line, the total intersection number is precisely the degree of $\Delta$.  The total intersection number is a sum (à la Fulton-MacPherson refined intersection theory) of positive integers (intersection multiplicities) for each connected component of $\Lambda \cap \Delta$.  
You could ask for a lower bound on the intersection multiplicity coming from the connected component (i.e., point) $f([1:0])$.  This would give a refined upper bound above.  This multiplicity should be computable via excess intersection.
