Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Density of rational points on del Pezzo surfaces of degree one (1212.2364v2)

Published 11 Dec 2012 in math.AG

Abstract: 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.

Summary

We haven't generated a summary for this paper yet.