I am interested in papers that investigates the existence or non-existence of curves of low degree (relative to the degree of the ambient variety). The starting example is that of surfaces and hypersurface of Fermat type, namely the variety $F = 0$ where

$$\displaystyle F(x_1, ..., x_n) = a_1 x_1^d + \cdots + a_n x_n^d$$

with $a_1, \cdots, a_n \in \mathbb{Z}$.

It was shown by Salberger and Marmon that the only curves of degree less than $(d+1)/3$ that lie on a Fermat surface when $n = 4$ are the 'trivial' lines. It can be shown independently, through methods in Thue equations for example, that these lines (while having low degree) have very few rational points on them. Thus, Fermat surfaces can be proved to have few rational points.

The result stating that Fermat surfaces essentially contains no curves of low degree is rather special and relies heavily on the shape of the equation. I am asking if there exists in the literature any results that attempt to tackle this problem for more general polynomials. Any input would be greatly appreciated.