Timeline for Rational points on surfaces of general type
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 9 at 15:57 | comment | added | Doug Liu | Dear @JasonStarr, May I ask why the set of rational points of a "variety of general type admitting a non-constant morphism to an Abelian variety" is not Zariski dense? | |
| Mar 8, 2013 at 5:31 | comment | added | Jérémy Blanc | Thanks Antoine for the comment and the reference, I downloaded the article and will read it. | |
| Mar 7, 2013 at 23:47 | comment | added | ACL | Dear Jeremy: I think this was the insight of Serge Lang to relate positivity properties of the canonical bundle, hyperbolicity and rational points in higher dimensions. He wrote a paper on that subject, projecteuclid.org/euclid.bams/1183553166 | |
| Mar 7, 2013 at 17:55 | comment | added | Jérémy Blanc | Thanks for the answer, I like it. However, I would like to have other results. For example if you take hypersurfaces of $\mathbb{P}^3$ of degree $d>4$, with singularities which are not too bad (so that it is of general type), can we say something? And apart from the proofs in some particular cases (which is the second question), is there a reason a priori why general type should imply non-Zariski dense rational points? (first question) | |
| Mar 7, 2013 at 16:13 | comment | added | Jason Starr | Even if the variety of general type just admits a non-constant morphism to an Abelian variety, that can be enough. Thus a variety that fibers a hyperbolic curve cannot have a Zariski dense set of rational points, since the rational points must lie in the finitely many fibers over the finitely many rational points of the base curve. | |
| Mar 7, 2013 at 13:50 | history | edited | ACL | CC BY-SA 3.0 |
typo; deleted 4 characters in body
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| Mar 7, 2013 at 13:50 | comment | added | ACL | In other words, Faltings's Theorem treats the case of varieties of general type which can be embedded in an Abelian variety. | |
| Mar 7, 2013 at 13:32 | history | answered | Siksek | CC BY-SA 3.0 |