Journal of the
Korean Mathematical Society JKMS

We show the existence of smooth isolated curves of different degrees and genera in Calabi-Yau threefolds that are complete intersections in homogeneous spaces. Along the way, we classify all degrees and genera of smooth curves on $BN$ general $K3$ surfaces of genus $\mu$, where $5 \leq \mu \leq 10$. By results of Mukai, these are the $K3$ surfaces that can be realised as complete intersections in certain homogeneous spaces.

Keywords: isolated curves, deformations, $BN$ general $K3$ surfaces, Calabi-Yau threefolds

MSC numbers: 14J32, 14J28