Bremner, Andrew On diagonal cubic surfaces. (English) Zbl 0652.14019 Manuscr. Math. 62, No. 1, 21-32 (1988). The author studies rational curves on the surface \(ax^3+by^3+cz^3=dt^3\) for cube-free rational numbers \(a,b,c,d\) extending work done by B. Segre [Math. Notae 11, 1–68 (1951; Zbl 0043.27501)] and H. P. F. Swinnerton-Dyer [1969 Number Theory Inst., Proc. Symp. Pure Math. 20, 1–52 (1971; Zbl 0228.14001)]. The author shows that there are no curves of arithmetic genus 0 defined over \(\mathbb Q\) unless (after permuting coordinates) \(a=b=1\) and \(c=d\). For the surface \(x^3+y^3- cz^3=ct^3\), he shows that there are exactly two such curves if \(c\neq 2\), and four if \(c=2\); and he gives explicit equations for these curves. Finally he generalizes an example of D. H. Lehmer [J. Lond. Math. Soc. 31, 275–280 (1956; Zbl 0072.26804)] to show that the surfaces \(x^3+y^3+cz^3=ct^3\) contain infinitely many curves defined over \(\mathbb Q\) of geometric genus 0 and arithmetic genus greater than 0. Reviewer: Joseph H. Silverman (Providence) Cited in 1 Document MSC: 11D25 Cubic and quartic Diophantine equations 11D72 Diophantine equations in many variables 11G35 Varieties over global fields 14G25 Global ground fields in algebraic geometry 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties Keywords:rational curves on cubic surface Citations:Zbl 0043.27501; Zbl 0228.14001; Zbl 0072.26804 PDFBibTeX XMLCite \textit{A. Bremner}, Manuscr. Math. 62, No. 1, 21--32 (1988; Zbl 0652.14019) Full Text: DOI EuDML References: [1] LEHMER, D.H., On the Diophantine equationx 3+y 3+z 3=1, J. London Math. Soc., 31, 275-280, (1956). · Zbl 0072.26804 · doi:10.1112/jlms/s1-31.3.275 [2] SEGRE, B., On the rational solutions of homogeneous cubic equations in four variables, Math. Notae (Rosario), 11, 1-68, (1951). · Zbl 0043.27501 [3] SWINNERTON-DYER, H. P. F., Applications of algebraic geometry to number theory, 1969 Institute on Number Theory, Amer. Math. Soc. Proceedings XX, 1-52. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.