Curves of Small Genus on Certain K3 Surfaces

Abstract

Recent workers [1, 3] have proved density theorems about the rational points on K3 surfaces of the form $$ V:\;X_0^4+cX_1^4=X_2^4+cX_3^4 $$ for certain non-zero values of c. Their arguments depend on the presence of at least two pencils of curves of genus 1 on V. Unfortunately the values of c for which the argument works are constrained by the need to exhibit explicitly a rational point on V which satisfies certain extra conditions; these in particular require it to lie outside the four obvious rational lines on V. It is therefore natural to ask whether there are other curves of genus 0 or 1 defined over Q on V. In the case c = 1 there are known to be infinitely many such curves (see [2]), and for general rational c the quadratic form Q on the Néron–Severi group whose value is the self-intersection number takes the values 0 and -2 infinitely often. Naively one might expect the case c = 1 to be typical; but this is not so. The main object of this paper is to prove the following result.