Isolated rational curves on K3-fibered Calabi-Yau threefolds

In this paper we study 16 complete intersection K3-fibered Calabi-Yau varie ty types in biprojective space P-n1 x P-1. These are all the CICY-types tha t are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d, 0) for every positive integer d on a general Calabi-Yau variety of these types. The proof depends heavily on existence theorems for curves on K3-surfaces proved by S. Mori and K. O guiso. Some of these varieties are related to Calabi-Yau varieties in proje ctive space by a determinantal contraction, and we use this to prove existe nce of rational curves of every degree for a general Calabi-Yau variety in projective space.