One-dimensional Local Families of Complex K3 Surfaces

For any complex K3 surface $X$, we construct a one-dimensional deformation in which all integers $\rho$ with $0\le \rho \le 20$ 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.