• Corpus ID: 104292609

Perfect points on genus one curves and consequences for supersingular K3 surfaces

@article{Bragg2019PerfectPO,
  title={Perfect points on genus one curves and consequences for supersingular K3 surfaces},
  author={Daniel Bragg and Max Lieblich},
  journal={arXiv: Algebraic Geometry},
  year={2019}
}
We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus one curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of such fibrations without purely inseparable multisections. Finally, we discuss the… 

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References

SHOWING 1-10 OF 26 REFERENCES
A Crystalline Torelli Theorem for Supersingular K3 Surfaces
I like to argue that crystalline cohomology will play a role in characteristic p analogous to the role of Ilodge theory in characteristic zero. One aspect of this analogy is that the F-crystal
On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic, II
Let $A$ be an abelian variety over the function field $K$ of a curve over a finite field. We describe several mild geometric conditions ensuring that the group $A(K^{\rm perf})$ is finitely generated
The Moduli of Weierstrass Fibrations Over IP 1
Let k be an algebraically closed field of characteristic 4 = 2, 3. Let X p , Y be a flat proper map of reduced irreducible k-schemes such that every geometric fibre is either (a) an elliptic curve,
On the Chow Groups of Supersingular Varieties
Abstract We compute the rational Chow groups of supersingular abelian varieties and some other related varieties, such as supersingular Fermat varieties and supersingular $K3$ surfaces. These
Supersingular K3 surfaces are unirational
We show that supersingular K3 surfaces in characteristic $$p\ge 5$$p≥5 are related by purely inseparable isogenies. This implies that they are unirational, which proves conjectures of Artin, Rudakov,
On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic
Let K be the function field of a smooth and proper curve S over an algebraically closed field k of characteristic p > 0. Let A be an ordinary abelian variety over K . Suppose that the N´eron model A of
Enriques surfaces with normal K3-like coverings
We analyze the structure of simply-connected Enriques surface in characteristic two whose K3-like covering is normal, building on the work of Ekedahl, Hyland and Shepherd-Barron. We develop general
On the unirationality of supersingular K3 surfaces
We prove that supersingular K3 surfaces over algebraically closed fields of characteristic at least $5$ are unirational, following a simplified form of Liedtke's strategy.
Twistor Spaces for Supersingular K3 Surfaces
We develop a theory of twistor spaces for supersingular K3 surfaces, extending the analogy between supersingular K3 surfaces and complex analytic K3 surfaces. Our twistor spaces are obtained as
Positivity of Hodge bundles of abelian varieties over some function fields
The main result of this paper concerns the positivity of the Hodge bundles of abelian varieties over global function fields. As applications, we obtain some partial results on the Tate–Shafarevich
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