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Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type

Benjamin Bakker, Andrei Jorza (2014)

Open Mathematics

We classify the cohomology classes of Lagrangian 4-planes ℙ4 in a smooth manifold X deformation equivalent to a Hilbert scheme of four points on a K3 surface, up to the monodromy action. Classically, the Mori cone of effective curves on a K3 surface S is generated by nonnegative classes C, for which (C, C) ≥ 0, and nodal classes C, for which (C, C) = −2; Hassett and Tschinkel conjecture that the Mori cone of a holomorphic symplectic variety X is similarly controlled by “nodal” classes C such that...

Local-global divisibility of rational points in some commutative algebraic groups

Roberto Dvornicich, Umberto Zannier (2001)

Bulletin de la Société Mathématique de France

Let 𝒜 be a commutative algebraic group defined over a number field  k . We consider the following question:Let r be a positive integer and let P ∈ 𝒜 ( k ) . Suppose that for all but a finite number of primes v of k , we have P = r D v for some D v ∈ 𝒜 ( k v ) . Can one conclude that there exists D ∈ 𝒜 ( k ) such that P = r D ?A complete answer for the case of the multiplicative group 𝔾 m is classical. We study other instances and in particular obtain an affirmative answer when r is a prime and  𝒜 is either an elliptic curve or a torus of small dimension...

Local-global principle for certain biquadratic normic bundles

Yang Cao, Yongqi Liang (2014)

Acta Arithmetica

Let X be a proper smooth variety having an affine open subset defined by the normic equation N k ( √ a , √ b ) / k ( x ) = Q ( t ₁ , . . . , t ₘ ) ² over a number field k. We prove that: (1) the failure of the local-global principle for zero-cycles is controlled by the Brauer group of X; (2) the analogue for rational points is also valid assuming Schinzel’s hypothesis.

Manin’s and Peyre’s conjectures on rational points and adelic mixing

Alex Gorodnik, François Maucourant, Hee Oh (2008)

Annales scientifiques de l'École Normale Supérieure

Let X be the wonderful compactification of a connected adjoint semisimple group G defined over a number field K . We prove Manin’s conjecture on the asymptotic (as T → ∞ ) of the number of K -rational points of X of height less than T , and give an explicit construction of a measure on X ( 𝔸 ) , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points 𝐆 ( K ) on X ( 𝔸 ) . Our approach is based on the mixing property of L 2 ( 𝐆 ( K ) ∖ 𝐆 ( 𝔸 ) ) which we obtain with a rate of convergence.

Manin’s conjecture for a singular sextic del Pezzo surface

Daniel Loughran (2010)

Journal de Théorie des Nombres de Bordeaux

We prove Manin’s conjecture for a del Pezzo surface of degree six which has one singularity of type A 2 . Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.

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