Lines on Fermat surfaces

Abstract

We prove that the Néron-Severi groups of several complex Fermat surfaces are generated by lines. Specifically, we obtain these new results for all degrees up to 100 that are relatively prime to 6. The proof uses reduction modulo a supersingular prime. The techniques are developed in detail. They can be applied to other surfaces and varieties as well.

Unit equations and Fermat surfaces in positive characteristic

“…We will also explain there why we need the condition that N is (480, p)-good. The rough reason is that if N is not (1, p)-good, then the Fermat surface is known to be unirational [13]. Our work shows that the unirationality of these surfaces is strongly related to the two-dimensional Frobenius families appearing in Theorem 1.…”

“…However, it is a more difficult task to find the smallest N using abc Theorems, see for example [3]. Our Theorem 4 is also based on abc type arguments and for this reason it should not be surprising that we can not distinguish between the case N = p r + 1, giving unirational surfaces [13], and N = ap r + b with 0 < a, b small.…”

“…Finally we will explain why we need the condition that N is (480, p)-good. If N = p r + 1 for some r ≥ 0, it is possible to write down non-trivial lines on the Fermat surface, see Section 5.1-5.4 of [13]. It turns out that our method is unable to distinguish between the case N = p r + 1 and N = ap r + b with 0 < a, b small.…”

On a smooth quartic surface containing 56 lines which is isomorphic as a K3 surface to the Fermat quartic

“…When Y is defined over C, the second cohomology group H 2 (Y, Z) of a complex K3 surface Y with the cup product is an even, unimodular lattice of signature (3,19) containing S Y as a primitive sublattice.…”

Picard numbers of quintic surfaces

“…In particular, we can compute the Picard number combinatorially by singling out all α ∈ A m whose Galois orbit does not leave the Hodge type (1, 1). Computations become especially transparent in special cases, for instance for degree m relatively prime to 6 where all of NS(S m ) is generated over Z by lines by [Deg13] (see also [Shi82] and [SSvL10]). …”