Brill-Noether general K3 surfaces with the maximal number of elliptic pencils of minimal degree

We explicitly construct Brill–Noether general $K3$ surfaces of genus $4,6$ and $8$ having the maximal number of elliptic pencils of degrees $3,4$ and $5$, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill–Noether general $K3$ surfaces of genus $4$ and $6$ without stable Lazarsfeld–Mukai bundles of minimal $c_2$.