Let $F$ and $G$ be coprime 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 to verify whether this pencil is isotrivial (general members of the pencil are isomorphic)?
I wish to avoid a brute force computation of possible isomorphisms in ${\rm PGL}(3,\mathbb{C})$.