[Paper Review] Perfect points on genus one curves and consequences for supersingular K3 surfaces

Daniel Bragg, Max Lieblich|arXiv (Cornell University)|Apr 9, 2019

Algebraic Geometry and Number Theory15 references4 citations

TL;DR

This paper establishes that for very general supersingular K3 surfaces in characteristic ≥5, no elliptic fibration admits a purely inseparable multisection unless it also has a rational section. Using Frobenius action on cohomology and singular fiber configurations in Weierstrass models, the authors prove that such multisections are impossible for Artin invariant 8, 9, or 10, leading to a strong restriction on rational points over perfect closures of function fields.

ABSTRACT

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 consequences for the claimed proof of the Artin conjecture on unirationality of supersingular K3 surfaces.

Motivation & Objective

• To determine when genus one curves over function fields admit points over the perfect closure, especially in the context of supersingular K3 surfaces.
• To resolve the existence of purely inseparable multisections on elliptic fibrations of supersingular K3 surfaces.
• To establish a cohomological criterion using Frobenius pullback on H²(O_J) to detect the absence of such multisections.
• To demonstrate that for very general supersingular K3 surfaces, purely inseparable multisections imply rational sections, thus linking arithmetic and geometric structures.

Proposed method

• Use Artin–Tate families to parametrize deformations of Brauer classes associated to non-Jacobian elliptic fibrations on supersingular K3 surfaces.
• Analyze the action of Frobenius on the tangent space of the Brauer class, identifying it with pullback on H²(O_J) for the Jacobian fibration.
• Apply a criterion based on singular fiber types (via Tate’s algorithm) to determine when the Frobenius pullback on cohomology is injective.
• Construct explicit Weierstrass models with specified Kodaira fiber types to verify the injectivity condition and rule out multisections.
• Use moduli spaces of Weierstrass data to bound the dimension of loci admitting purely inseparable multisections.
• Leverage the non-∞-Frobenius split locus to show that general fibrations fail to admit such multisections.

Experimental results

Research questions

• RQ1 Under what conditions does a genus one curve over k(t) admit a point over the perfect closure k(t)perf?
• RQ2 Can purely inseparable multisections exist on elliptic fibrations of supersingular K3 surfaces in characteristic ≥5?
• RQ3 Is the existence of a purely inseparable multisection equivalent to the existence of a rational section for very general supersingular K3 surfaces?
• RQ4 How do singular fiber configurations in Weierstrass models affect the injectivity of Frobenius pullback on H²(O_J)?
• RQ5 What is the dimension of the locus of Weierstrass models admitting purely inseparable multisections?

Key findings

• For very general supersingular K3 surfaces of Artin invariant 8, 9, or 10, purely inseparable multisections exist if and only if rational sections exist.
• No elliptic fibration on a very general supersingular K3 surface of Artin invariant 10 admits a purely inseparable multisection, due to the absence of rational sections.
• The Frobenius pullback on H²(O_J) is injective for certain configurations of singular fibers, such as those with multiple III, IV, or I_n^* fibers, which prevents the Brauer class from vanishing.
• Explicit examples of supersingular K3 surfaces without purely inseparable multisections are constructed via Weierstrass models with specific fiber type combinations.
• The locus of Weierstrass data admitting purely inseparable multisections is of codimension ≥1 in the moduli space, implying generically no such multisections exist.
• The non-∞-Frobenius split locus in WD^{-2} is non-empty and contains configurations where the Frobenius pullback fails to split, supporting the non-existence result.