An afterthought on the generalized Mordell-Lang conjecture
نویسنده
چکیده
The generalized Mordell-Lang conjecture (GML) is the statement that the irreducible components of the Zariski closure of a subset of a group of finite rank inside a semi-abelian variety are translates of closed algebraic subgroups. In [6], M. McQuillan gave a proof of this statement. We revisit his proof, indicating some simplifications. This text contains a complete elementary proof of the fact that (GML) for groups of torsion points (= generalized Manin-Mumford conjecture), together with (GML) for finitely generated groups imply the full generalized Mordell-Lang conjecture. Mathematics Subject Classification: 14G05, 14K15
منابع مشابه
A Combination of the Conjectures of Mordell-Lang and André-Oort
We propose a conjecture combining the Mordell-Lang conjecture with an important special case of the André-Oort conjecture, and explain how existing results imply evidence for it. Mathematics Subject Classification: 14G35, 14K12 (11F32, 11F72, 11G15, 11G18)
متن کاملThe Mordell-lang Conjecture in Positive Characteristic Revisited
We prove versions of the Mordell-Lang conjecture for semiabelian varieties defined over fields of positive characteristic.
متن کاملAbelian Varieties and the Mordell–Lang Conjecture
This is an introductory exposition to background material useful to appreciate various formulations of the Mordell–Lang conjecture (now established by recent spectacular work due to Vojta, Faltings, Hrushovski, Buium, Voloch, and others). It gives an exposition of some of the elementary and standard constructions of algebro-geometric models (rather than model-theoretic ones) with applications (...
متن کاملOn a Dynamical Mordell-lang Conjecture for Coherent Sheaves
We introduce a dynamical Mordell-Lang-type conjecture for coherent sheaves. When the sheaves are structure sheaves of closed subschemes, our conjecture becomes a statement about unlikely intersections. We prove an analogue of this conjecture for affinoid spaces, which we then use to prove our conjecture in the case of surfaces. These results rely on a module-theoretic variant of Strassman’s the...
متن کاملOn a Uniform Bound for the Number of Exceptional Linear Subvarieties in the Dynamical Mordell–lang Conjecture
Let φ : P → P be a morphism of degree d ≥ 2 defined over C. The dynamical Mordell–Lang conjecture says that the intersection of an orbit Oφ(P ) and a subvariety X ⊂ P is usually finite. We consider the number of linear subvarieties L ⊂ P such that the intersection Oφ(P ) ∩ L is “larger than expected.” When φ is the d-power map and the coordinates of P are multiplicatively independent, we prove ...
متن کامل