Rational curves on elliptic K3 surfaces

Abstract

Given a Brauer class on a K3 surface defined over a number field, we prove that there exists infinitely many specializations where the Brauer class vanishes, under certain technical hypotheses, answering a question of Frei-Hassett-Várilly-Alvarado. Contents 1. Introduction 1 2. K3 surfaces and Brauer classes: some reductions 3 3. GSpin Shimura varieties: integral models and Arakelov intersection theory 6 4. Global estimate 16 5. Archimedean estimates 23 6. Non-archimedean estimates 27 References 28

Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

“…Otherwise, there is one specialization of residual charactertic different from 2 and 3 and which is supersingular, hence admits an elliptic firation by the Tate conjecture. By [Tay18b], such specialization admits geometrically infinitely many rational curves and we conclude by Proposition 5.1 in [CGL19]. 8.2.…”

Singular rational curves on elliptic K3 surfaces

“…The method of the proof of Theorem 1.1 builds on the techniques by Bogomolov–Tschinkel [1] and Hassett [5], who constructed infinitely many rational curves on a complex elliptic K3 surface by using the multiplication map of the elliptic structure. Their results have since been extended to characteristic $$p>3$$ by Tayou in [9]. The main idea is to start with a rational curve R and look at its image under the rational multiplication map.…”

Singular Rational Curves on Elliptic K3 Surfaces

“…The method of the proof of Theorem 1.1 builds on the techniques by Bogomolov-Tschinkel [1] and Hassett [5] who constructed infinitely many rational curves on a complex elliptic K3 surface. Their results have since also been extended to characteristic p > 3 by Tayou in [9]. The main idea is to start with a rational curve R and look at its image under certain rational maps between elliptic K3 surfaces.…”