Rational points on certain del Pezzo surfaces of degree one

2009/01/17 by Ulas, Maciej · 1 citation
#11D25 #11D41 #11G052 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0901.2658

Abstract

Let f(z)=z⁵+az³+bz²+cz+d ∈ \Z[z] and let us consider a del Pezzo surface of degree one given by the equation \calE_f: x²-y³-f(z)=0. In this note we prove that if the set of rational points on the curve E_{a, b}:Y²=X³+135(2a-15)X-1350(5a+2b-26) is infinite, then the set of rational points on the surface \calE_f is dense in the Zariski topology.

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• Hilbert property for low-genus families and degree-one del Pezzo surfaces

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