472 research outputs found
Hasse principle for Kummer varieties in the case of generic 2-torsion
Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and
Skorobogatov used Swinnerton-Dyer's descent-fibration method to establish the
Hasse principle for Kummer varieties associated to a 2-covering of a
principally polarised abelian variety under certain largeness assumptions on
its mod 2 Galois image. Their method breaks down however when the Galois image
is maximal, due to the possible failure of the Shafarevich--Tate group of
quadratic twists of A to have square order. In this work we overcome this
obstruction by combining second descent ideas in the spirit of Harpaz and Smith
with results on the parity of 2-infinity Selmer ranks in quadratic twist
families. This allows Swinnerton-Dyer's method to be successfully applied to K3
surfaces arising as quotients of 2-coverings of Jacobians of genus 2 curves
with no rational Weierstrass points.Comment: 23 pages, comments welcom
Supersingular K3 Surfaces are Unirational
We show that supersingular K3 surfaces in characteristic are related
by purely inseparable isogenies. This implies that they are unirational, which
proves conjectures of Artin, Rudakov, Shafarevich, and Shioda. As a byproduct,
we exhibit the moduli space of rigidified K3 crystals as an iterated
-bundle over . To complete the picture, we also
establish Shioda-Inose type isogeny theorems for K3 surfaces with Picard rank
in positive characteristic.Comment: 31 pages; many details added, final versio
Lagrangian fibrations of holomorphic-symplectic varieties of K3^[n]-type
Let X be a compact Kahler holomorphic-symplectic manifold, which is
deformation equivalent to the Hilbert scheme of length n subschemes of a K3
surface. Let L be a nef line-bundle on X, such that the 2n-th power of c_1(L)
vanishes and c_1(L) is primitive. Assume that the two dimensional subspace
H^{2,0}(X) + H^{0,2}(X), of the second cohomology of X with complex
coefficients, intersects trivially the integral cohomology. We prove that the
linear system of L is base point free and it induces a Lagrangian fibration on
X. In particular, the line-bundle L is effective. A determination of the
semi-group of effective divisor classes on X follows, when X is projective. For
a generic such pair (X,L), not necessarily projective, we show that X is
bimeromorphic to a Tate-Shafarevich twist of a moduli space of stable torsion
sheaves, each with pure one dimensional support, on a projective K3 surface.Comment: 34 pages. v3: Reference [Mat5] and Remark 1.8 added. Incorporated
improvement to the exposition and corrected typos according to the referees
suggestions. To appear in the proceedings of the conference Algebraic and
Complex Geometry, Hannover 201
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