(Non-)Existence of curves of low degree on affine and projective varieties 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.
 A: You may already know this but there is this really fun paper by Bruce Reznick where he finds some interesting rational curves on Fermat hypersurfaces. For example conics on the degree 5 Fermat hypersurface in P^3. See Matt Deland's thesis for more on these examples (page 52 ff) and more on rational curves on hypersurfaces.
As was mentioned in the comments, algebraic geometers working over algebraically closed fields know very little for any particular hypersurface and quite a bit (part of it conjectural) for a general hypersurface. Hope you can help us!
A: There is a huge literature about bounding the rational curves on general type hypersurfaces, both general and special.  Just to start the list: Y.-T. Siu, Michael McQuillan, Xu, Clemens, Ein, Voisin, Pacienza, ... 
A: For curves of degree one you could look at Section 2.4 of Debarre's book "Higher dimensional algebraic geometry". Section 2.14 for Fermat hypersurfaces. 
More generally for low degree rational curves:
For hypersurfaces: http://arxiv.org/abs/math/0203088
For conic connected varieties: http://arxiv.org/abs/math/0701885
For conics in complete intersections: http://arxiv.org/abs/0804.1627
For arbitrary projective varieties: http://arxiv.org/abs/1106.0124
