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!



Number of singular fibers in families of hypersurfaces
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](https://mathoverflow.net/questions/165672/bounding-the-number-of-critical-points-in-a-lefschetz-pencil/165674#165674) (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*.


