Density of rational points on del Pezzo surfaces of degree one

We state conditions under which the set S(k) of k-rational points on a del Pezzo surface S of degree 1 over an infinite field k of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic fibration over the projective line induced by the anticanonical map has a nodal fiber over a k-rational point. It also suffices to require the existence of a point in S(k) that does not lie on six exceptional curves of S and that has order 3 on its fiber of the elliptic fibration. This allows us to show that within a parameter space for del Pezzo surfaces of degree 1 over the field of real numbers, the set of surfaces S defined over the field Q of rational numbers for which the set S(Q) is Zariski dense, is dense with respect to the real analytic topology. We also include conditions that may be satisfied for every del Pezzo surface S and that can be verified with a finite computation for any del Pezzo surface S that does satisfy them.Comment: 31 pages; the main results have not changed; the presentation has been improved; a magma file that checks all computations may be obtained from arXiv by downloading the source of this articl

Unirationality of del Pezzo surfaces of degree two over finite fields

We prove that every del Pezzo surface of degree two over a finite field is unirational, building on the work of Manin and an extension by Salgado, Testa, and V\'arilly-Alvarado, who had proved this for all but three surfaces. Over general fields of characteristic not equal to two, we state sufficient conditions for a del Pezzo surface of degree two to be unirational.Comment: This is a short version (5 pages) of arXiv:1408.0269; the longer version contains more details and more general result

Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces

We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler's work on the second problem with a modern point of view.Comment: 11 pages, 1 figur