Picard numbers in a family of hyperk\"ahler manifolds - A supplement to the article of R. Borcherds, L. Katzarkov, T. Pantev, N. I. Shepherd-Barron

TL;DR

This paper investigates how the Picard number of hyperk"ahler manifolds varies under small deformations, highlighting the density of jumping loci and exploring implications for Mordell-Weil groups of Jacobian K3 surfaces.

Contribution

It provides new insights into the density of Picard number jumps in hyperk"ahler families and applies these results to the study of Mordell-Weil groups of Jacobian K3 surfaces.

Findings

Density of Picard number jumping loci under small deformations

Applications to Mordell-Weil groups of Jacobian K3 surfaces

Enhanced understanding of Picard number variation in hyperk"ahler manifolds

Abstract

We remark the density of the jumping loci of the Picard number of a hyperk\"ahler manifold under small one-dimensional deformation and provide some applications for the Mordell-Weil groups of Jacobian K3 surfaces.