One-dimensional Local Families of Complex K3 Surfaces

TL;DR

This paper constructs a one-dimensional family of complex K3 surfaces where all Picard numbers from 0 to 20 appear, contrasting with the generic case where only 0 and 1 occur.

Contribution

It introduces a specific one-dimensional deformation of K3 surfaces that realizes all Picard numbers from 0 to 20, expanding understanding of Picard number variations.

Findings

Constructed a deformation with all Picard numbers 0 to 20

Proved generic families only have Picard numbers 0 and 1

Highlighted differences between special and generic families

Abstract

For any complex K3 surface $X$, we construct a one-dimensional deformation in which all integers $\rho$ with $0 \leq \rho \leq 20$ occur as Picard numbers of some fibres. In contrast, we prove that the generic one-dimensional local family of K3 surfaces admits only $0$ and $1$ as Picard numbers of the fibres.