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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.

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    $\begingroup$ The Noether Lefschetz theorem says that the Picard group of a very general surface of degree $d\ge 4$ of $\mathbb P^3$ is cyclic generated by the hyperplane section, so in this case all curves have degree divisible by $d$. ("Very general" means "in the complement of a countable union of Zariski closed subsets"). $\endgroup$
    – rita
    Commented Dec 29, 2013 at 11:44
  • $\begingroup$ You mention Thue and Salberger. Are you working over an algebraically closed field? Are you working over a number field? If so, which number field? $\endgroup$
    – Jason Starr
    Commented Dec 30, 2013 at 12:46
  • $\begingroup$ Salberger's result concerns Fermat surfaces over algebraically closed fields (which in turn implies that no curves of low degree can exist over any other number field) $\endgroup$
    – Stanley Yao Xiao
    Commented Dec 30, 2013 at 13:10
  • $\begingroup$ And the methods in Thue equations concerns results dealing with the Thue equation $ax^d + by^d = h$ for $d \geq 3$, which was shown by Evertse to have $O(h^\epsilon)$ many solutions. $\endgroup$
    – Stanley Yao Xiao
    Commented Dec 30, 2013 at 13:56

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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!

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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, ...

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  • $\begingroup$ What would you recommend as to how to delve into this literature? Like which papers are the most authoritative/important? $\endgroup$
    – Stanley Yao Xiao
    Commented Jan 2, 2014 at 2:05
  • $\begingroup$ There is a nice survey article about hyperbolicity by Izzet Coskun on his webpage. That might be a good place to start. Also the article by Pacienza has a nice survey of the literature up to the date of his article. $\endgroup$
    – Jason Starr
    Commented Jan 2, 2014 at 2:21
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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

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