Matches in SemOpenAlex for { <https://semopenalex.org/work/W2008047689> ?p ?o ?g. }
Showing items 1 to 85 of
85
with 100 items per page.
- W2008047689 abstract "Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. A Mordell-Weil generating set is a subset B of S(K) of minimal cardinality which generates S(K) via successive secant and tangent constructions. Let r(S,K) be the cardinality of such a Mordell-Weil generating set. Manin posed what is known as the Mordell-Weil problem for cubic surfaces: if K is finitely generated over its prime subfield then r(S,K) is finite. In this paper, we prove a special case of this conjecture. Namely, if S contains two skew lines both defined over K then r(S,K) = 1. One of the difficulties in studying the secant and tangent process on cubic surfaces is that it does not lead to an associative binary operation as in the case of elliptic curves. As a partial remedy we introduce an abelian group H_S(K) associated to a cubic surface S/K, naturally generated by the K-rational points, which retains much information about the secant and tangent process. In particular, r(S, K) is large as soon as the minimal number of generators of H_S(K) is large. In situations where weak approximation holds, H_S has nice local-to-global properties. We use these to construct a family of smooth cubic surfaces over the rationals such that r(S,K) is unbounded in this family. This is the cubic surface analogue of the unboundedness of ranks conjecture for elliptic curves." @default.
- W2008047689 created "2016-06-24" @default.
- W2008047689 creator A5017561176 @default.
- W2008047689 date "2010-12-08" @default.
- W2008047689 modified "2023-09-27" @default.
- W2008047689 title "On the number of Mordell-Weil generators for cubic surfaces" @default.
- W2008047689 cites W1493505561 @default.
- W2008047689 cites W1596671514 @default.
- W2008047689 cites W1970793402 @default.
- W2008047689 cites W2016257935 @default.
- W2008047689 cites W2081741202 @default.
- W2008047689 cites W2141049063 @default.
- W2008047689 cites W2314530875 @default.
- W2008047689 cites W2319423512 @default.
- W2008047689 cites W3141719238 @default.
- W2008047689 cites W2796701678 @default.
- W2008047689 hasPublicationYear "2010" @default.
- W2008047689 type Work @default.
- W2008047689 sameAs 2008047689 @default.
- W2008047689 citedByCount "0" @default.
- W2008047689 crossrefType "posted-content" @default.
- W2008047689 hasAuthorship W2008047689A5017561176 @default.
- W2008047689 hasConcept C114614502 @default.
- W2008047689 hasConcept C118615104 @default.
- W2008047689 hasConcept C124101348 @default.
- W2008047689 hasConcept C134306372 @default.
- W2008047689 hasConcept C136170076 @default.
- W2008047689 hasConcept C138187205 @default.
- W2008047689 hasConcept C156330621 @default.
- W2008047689 hasConcept C179603306 @default.
- W2008047689 hasConcept C194254661 @default.
- W2008047689 hasConcept C202444582 @default.
- W2008047689 hasConcept C2524010 @default.
- W2008047689 hasConcept C2780990831 @default.
- W2008047689 hasConcept C33923547 @default.
- W2008047689 hasConcept C41008148 @default.
- W2008047689 hasConcept C46875033 @default.
- W2008047689 hasConcept C87117476 @default.
- W2008047689 hasConcept C91429207 @default.
- W2008047689 hasConcept C9376300 @default.
- W2008047689 hasConceptScore W2008047689C114614502 @default.
- W2008047689 hasConceptScore W2008047689C118615104 @default.
- W2008047689 hasConceptScore W2008047689C124101348 @default.
- W2008047689 hasConceptScore W2008047689C134306372 @default.
- W2008047689 hasConceptScore W2008047689C136170076 @default.
- W2008047689 hasConceptScore W2008047689C138187205 @default.
- W2008047689 hasConceptScore W2008047689C156330621 @default.
- W2008047689 hasConceptScore W2008047689C179603306 @default.
- W2008047689 hasConceptScore W2008047689C194254661 @default.
- W2008047689 hasConceptScore W2008047689C202444582 @default.
- W2008047689 hasConceptScore W2008047689C2524010 @default.
- W2008047689 hasConceptScore W2008047689C2780990831 @default.
- W2008047689 hasConceptScore W2008047689C33923547 @default.
- W2008047689 hasConceptScore W2008047689C41008148 @default.
- W2008047689 hasConceptScore W2008047689C46875033 @default.
- W2008047689 hasConceptScore W2008047689C87117476 @default.
- W2008047689 hasConceptScore W2008047689C91429207 @default.
- W2008047689 hasConceptScore W2008047689C9376300 @default.
- W2008047689 hasLocation W20080476891 @default.
- W2008047689 hasOpenAccess W2008047689 @default.
- W2008047689 hasPrimaryLocation W20080476891 @default.
- W2008047689 hasRelatedWork W1501800212 @default.
- W2008047689 hasRelatedWork W1697297370 @default.
- W2008047689 hasRelatedWork W2001247847 @default.
- W2008047689 hasRelatedWork W2008345563 @default.
- W2008047689 hasRelatedWork W2051673319 @default.
- W2008047689 hasRelatedWork W2057500645 @default.
- W2008047689 hasRelatedWork W2269083353 @default.
- W2008047689 hasRelatedWork W2816327657 @default.
- W2008047689 hasRelatedWork W2884240507 @default.
- W2008047689 hasRelatedWork W2889752289 @default.
- W2008047689 hasRelatedWork W2938700811 @default.
- W2008047689 hasRelatedWork W2963812675 @default.
- W2008047689 hasRelatedWork W2965301522 @default.
- W2008047689 hasRelatedWork W2985205564 @default.
- W2008047689 hasRelatedWork W3037944244 @default.
- W2008047689 hasRelatedWork W3093228584 @default.
- W2008047689 hasRelatedWork W3103816784 @default.
- W2008047689 hasRelatedWork W3105407227 @default.
- W2008047689 hasRelatedWork W3658488 @default.
- W2008047689 hasRelatedWork W174794192 @default.
- W2008047689 isParatext "false" @default.
- W2008047689 isRetracted "false" @default.
- W2008047689 magId "2008047689" @default.
- W2008047689 workType "article" @default.