Abstract
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 <TEX>${\mu}$</TEX>, where <TEX>$5{\leq}{\mu}{\leq}10$</TEX>. By results of Mukai, these are the K3 surfaces that can be realised as complete intersections in certain homogeneous spaces.
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