To describe curves of form $C_{f,g}\stackrel{\mathrm{def}}{=}\left\{(x,y)\left|f\right(x)-g(y)=0\right\}$ and their number theory properties, you must address $C_{f,g}$ whose projective normalization has a $\mathrm{genus} 0 (\mathrm{or} 1)$ component. For $f$ and $g$ polynomials and $f$ indecomposable, [Fr73a] distinguished $C_{f,g}$ with $u=1$ versus $u>1$ components (Schinzel’s problem). For $u=1$, [Fr73b, (1.6) of Prop. 1] gave a direct genus formula. To complete $u>1$ required an adhoc genus computation.

[Fr12] revisited later work. Pakovich [Pak18b], an example, dropped the indecomposable and polynomial restrictions, but added $C_{f,g}$ is irreducible ($u=1$). He showed – for fixed $f$ – unless the Galois closure of the cover for $f$ has $\mathrm{genus} 0 (\mathrm{or} 1)$, the genus grows linearly in deg($g$). Cor. 2.20 and Cor. 2.21 extend [Fr73b, Prop. 1] and use Nielsen classes to generalize Pakovich’s formulation for $u>1$.

Using the solution to the Davenport and Schinzel problems, Hurwitz families track the significance of these components, an approach motivated by Riemann’s relating $\theta$ functions and half-canonical classes.