  To figure properties of a curve of form $C_{f,g} = {(x,y)| f(x) - g(y)= 0}$
you must address the genus 0 and 1 components of its projective normalization
$\tilde C_{f,g}$. For $f$ and $g$ polynomials with $f$ indecomposable, [Fr73a]
distinguished $\tilde C_{f,g}$ with $u=1$ versus $u > 1$ components (Schinzel's
problem). For $u = 1$, [Prop. 1, Fr73b] gave a direct genus formula. To
complete $u > 1$ required an adhoc genus computation.
  [Pak22] dropped the indecomposable and polynomial restrictions but added
$\tilde C_{f,g}$ is irreducible ($u = 1$). He showed - for fixed $f$ - unless
the Galois closure of the cover for $f$ has genus 0 or 1, the genus grows
linearly in deg($g$). Method I and Method II extend [Prop. 1, Fr73b}] using
Nielsen classes to generalize Pakovich's formulation for $u > 1$.
  Method I plays on the covers $f$ and $g$ to the $z$-line, $P^1_z$, from which
we compute the fiber product.
  Method II uses the projection to the $y$-line, $P^1_y$, based on explicitly
computing branch cycles for this cover.
  Hurwitz families track the significance of these components. Expanding on
[Prop. 2, Fr73a] shows how to approach Pakovich's problem. With no loss, start
with ($f^*,g^*$) which have the same Galois closures, and for which their
canonical representations are entangled. They, therefore, produce more than one
component on the fiber product.
  Then, we classify the possible component types, $W$, that appear on $\tilde
C_{f^*,g^*}$ using the branch cycles for $W$ that come from Method II. The
result is a Nielsen class formulation telling explicitly what $g_1\,$s to avoid
to assure the growth of the component genuses of $\tilde C_{f*,g*og_1}$ as
deg($g_1$) increases. Of particular note: using and expanding on Nielsen
classes and the solution of the genus 0 problem (classifying the monodromy
groups of indecomposable rational functions).
