About the isotriviality of pencils of plane curves Let $F$ and $G$ be coprime complex homogeneous polynomials in three variables of the same degree $d\geq 4$. Suppose that a general member of the pencil $\{F+tG=0\}\subset \mathbb{P}^2$ is smooth.
Which are the tools one can use to verify whether this pencil is isotrivial?
 A: This is not an answer but rather a lengthy comment.
A necessary condition for the pencil to be isotrivial is that a smooth member in that pencil has a non-trivial automorphism: By blowing-up the base points, $\mathbf{P}^2$ is birational to the total space of an isotrivial family $f : X \to \mathbf{P}^1$ of curves of genus $\ge 2$. Let $C$ denote a smooth fiber of $f$.
As $f$ is isotrivial, there exists a Galois cover $B \to \mathbf{P}^1$ (with the Galois group denoted by $G$) and a $G$-action on $C$ such that $f$ is birational to the projection $(C \times B)/G \to \mathbf{P}^1$. If $G$ acts trivially on $C$, then $\mathbf{P}^2$ would be birational to $C \times \mathbf{P}^1$, which is not possible.
This implies in particular that isotrivial pencils on $\mathbf{P}^2$ cannot be Lefschetz, 
because the automorphism group of a general member of a non-hyperelliptic Lefschetz pencil on a simply connected surface is trivial (see 10.6.18 in the book of Katz and Sarnak for a monodromy argument proving this fact). 
